Answer:
The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units
Step-by-step explanation:
In calculating the sample size to estimate a population proportion in which there is no information on an initial sample proportion, the principle of maximum uncertainty is assumed and a ratio [tex]P = 1/2[/tex] is assumed. The expression to calculate the size is:
[tex] n=\frac{z_{\alpha /2}^2}{4 \epsilon^2} [/tex]
With
Z value (for 0.005) [tex] Z _ {\alpha / 2} = 1.64485 [/tex]
Significance level [tex] \alpha = 0.01 [/tex]
Estimation error [tex] \epsilon = 0.03 [/tex]
[tex] n=\frac{(1.64485)^2}{(4)(0.03)^2} = 751.5398[/tex]
The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units
To determine the percentage of male customers with a margin of error of 3% at a 99% confidence level, the manager would need to survey approximately 1847 customers.
Explanation:The question is related to the concept of statistics, and more specifically to the idea of a confidence interval
for a population proportion. When you want to be very sure about your estimates, you use a high level of confidence. The standard formula for calculating the sample size needed in order to get a certain margin of error at a certain confidence level is n = [Z^2 * P * (1-P)] / E^2. In this formula, n is the sample size, Z is the z-score associated with your desired level of confidence, P is the preliminary estimate of the population proportion, and E is the desired margin of error. If the manager doesn't have a precursory idea of what the proportion of male customers is, it's standard to use P = 0.5. The Z score for a 99% confidence interval is approximately 2.576. Substituting these values, the manager would need a sample size of approximately 1846 customers. For more accuracy, it's better to round up to the next nearest whole number, so the minimum sample size required would be 1847.
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Please help me with geometry. Theres only 3 questions i need help with
If f(x) = 2x + 6 and g(x) = x, what is (gºf)(0)?
Final answer:
To find (g°f)(0), evaluate f(0) to get 6, then apply g to this result to also get 6. Hence, (g°f)(0) equals 6.
Explanation:
The question is asking to evaluate the composition of two functions, (g°f)(x), at the value x = 0. Composition of functions involves applying one function to the result of another. In this case, since f(x) = 2x + 6 and g(x) = x, we first find f(0), which is the result of plugging 0 into f(x).
Let's calculate:
f(0) = 2(0) + 6 = 6.
Next, we apply g to this result:
g(f(0)) = g(6) = 6.
Therefore, (g°f)(0) = 6.
The 1992 world speed record for a bicycle (human-powered vehicle) was set by Chris Huber. His time through the measured 200 m stretch was a sizzling 6.509 s, at which he commented,"Cogito ergo zoom!" (I think, therefore I go fast!). a.) What was Chris Huber’s speed in meters per second(m/s)? b) In 2001, Sam Whittingham beat Huber’s record by 19.0 km/h. What was Whittingham’s time through the 200 m? (answer hours)
Answer:
a) 30.726m/s and b) 5.5549s
Step-by-step explanation:
a.) What was Chris Huber’s speed in meters per second(m/s)?
Given the distance and time, the formula to obtain the speed is
[tex]v=\frac{d}{t}[/tex].
Applying this to our problem we have that
[tex]v=\frac{200m}{6.509s}= 30.726m/s[/tex].
So, Chris Huber’s speed in meters per second(m/s) was 30.726m/s.
b) What was Whittingham’s time through the 200 m?
In a) we stated that [tex]v=\frac{d}{t}[/tex]. This formula implies that
[tex]t=\frac{d}{v}[/tex].First, observer that [tex]19\frac{km}{h} =19,000\frac{m}{h}=\frac{19,000}{3,600}m/s= 5.2777m/s[/tex].
Then, Sam Whittingham speed was equal to Chris Huber’s speed plus 5.2777 m/s. So, [tex]v=30.726\frac{m}{s} +5.2777\frac{m}{s}= 36.003 m/s.[/tex]
Then, applying 1) we have that
[tex]t=\frac{200m}{36.003m/s}=5.5549s.[/tex]
So, Sam Whittingham’s time through the 200 m was 5.5549s.
Determine the sum of the first k odd positive integers for a number of values of k. What generalizations occur to you? Are your inferences correct for all positive integers k?
Answer:
[tex]S_{n} = \sum_{k=1}^{n} (2k-1) = n^2[/tex]
Step-by-step explanation:
Let's take a look at the first few odd numbers and their sum.
Lets define [tex]O_{k}[/tex] as the [tex]kth[/tex] odd number as:
[tex]O_{k} = 2k-1[/tex]
So we have:
[tex]O_{1} = 1\\O_{2} = 3\\O_{3} = 5\\O_{4} = 7\\[/tex]
And lets define the sum of all the odd numbers from [tex]k=1[/tex] to [tex]k=n[/tex] as:
[tex]S_{n} = \sum_{k=1}^n O_{k} = \sum_{k=1}^n (2k-1)[/tex]
Lets now check some values of said sum:
[tex]S_{1} = 1\\S_{2} = 1 + 3 = 4\\S_{3} = 1 + 3 + 5 = 9\\S_{4} = 1+3+5+7 = 16[/tex]
We can then observe than the sum up to [tex]n[/tex] equals [tex]n^2[/tex]
Let us then prove that this is the case by Induction.
First of all, we can prove this by an Induction Proof because we are taking all positive Integers. This is, we are working with the set of natural numbers [tex]\mathbb{N}[/tex].
We want to prove that
[tex]P(n) = S_{n} = \sum_{k=1}^n = n^2 \forall n\in \mathbb{N}[/tex]
This is, we want to prove that the sum of all odd numbers from [tex]1[/tex] to [tex]n[/tex] equals [tex]n^2[/tex] for all natural numbers.
Now, in order to prove something by Induction we need to check 2 things:
[tex]1) The\ base\ case . \ The\ statement\ holds\ for\ n=1\\2) The\ inductive\ step.\ Prove\ that\ if\ the\ statement\ holds\ for\ n\ then\ it\ must\ hold\ for\ n+1\\[/tex]
[tex]P(1)[/tex] is immediate:
[tex]P(1) = \sum_{k=1}^1 2k-1 = 1 = 1^2[/tex]
Now let's assume the statement holds for [tex]P(n)[/tex] and let's take a look at [tex]P(n+1)[/tex]
[tex]P(n+1) = \sum_{k=1}^{n+1} 2k-1[/tex]
And we can rewrite it by taking the last term out as:
[tex]P(n+1) = \sum_{k=1}^n 2k-1 \ + 2.(n+1) - 1[/tex]
And by inductive hypothesis we know that [tex]\sum_{k=1}^n 2k-1 = n^2[/tex]
and then:
[tex]P(n+1) = \sum_{k=1}^n 2k-1 \ + 2.(n+1) -1 = n^2 + 2n +2 -1 = n^2 +2n +1 = (n+1)^2[/tex]
And we have the proof we were looking for!
A hospital claims that the proportion, p, of full-term babies born in their hospital that weigh more than 7 pounds is 36%. In a random sample of 170 babies born in this hospital, 56 weighed over 7 pounds. Is there enough evidence to reject the hospital's claim at the level of significance?
Answer:
Claim :The proportion, p, of full-term babies born in their hospital that weigh more than 7 pounds is 36%.
n = 170
x = 56
We will use one sample proportion test
[tex]\widehat{p}=\frac{x}{n}[/tex]
[tex]\widehat{p}=\frac{56}{170}[/tex]
[tex]\widehat{p}=0.3294[/tex]
The proportion, p, of full-term babies born in their hospital that weigh more than 7 pounds is 36%.
[tex]H_0:p \neq 0.36 \\H_a:p= 0.36[/tex]
Formula of test statistic =[tex]\frac{\widehat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
[tex]=\frac{0.3294-0.36}{\sqrt{\frac{0.36(1-0.36)}{170}}}[/tex]
=−0.8311
Now refer the p value from the z table
P-Value is .202987 (Calculated by online calculator)
Level of significance α = 0.05
Since p value < α
So we reject the null hypothesis .
Hence the claim is true
You're driving into Canada and trying to decide whether to fill your gas tank before or after crossing the border. Gas in the United States costs $2.58/gallon, in Canada it's $1.29/L, and the Canadian dollar is worth 79¢ in U.S. currency. Where you should fill up?
Answer:
It is more convenient to fill up in the United States.
Step-by-step explanation:
We convert a US gallon to liters:
1 Gallon equals 3.78541 liters.
Therefore, 1 US Gallon costs (3.78541) x (1.29) = 4.8831789 Canadian dollars.
Now we convert the price of a US gallon in Canadian territory to US dollars:
4.8831789 * 0.79 = 3.85771133 US dollars.
Conclusion: A gallon purchased in the United States costs 2.58 US dollars, while a gallon in Canada is equivalent to 3.85771133 US dollars. This way it is more convenient to fill up in the United States.
It would be more economical to fill up the gas tank in the United States before crossing into Canada.
To determine whether it is more cost-effective to fill up the gas tank in the United States or Canada, we need to convert the Canadian gas price into U.S. dollars per gallon. The price of gas in Canada is $1.29 per liter. Since 1 gallon is equal to 3.78541 liters, the price per gallon in Canada would be $1.29 × 3.78541 = $4.88238 CAD. Now, we need to convert this price into USD, knowing the Canadian dollar is worth 79¢ in U.S. currency. Therefore, $4.88238 CAD × 0.79 = $3.85728 USD per gallon.
The price of gas in the United States is $2.58 per gallon. When comparing the two prices, it is clear that $2.58 per gallon in the United States is cheaper than the converted price of gas in Canada ($3.85728 per gallon in USD). Hence, it would be more economical to fill up the gas tank in the United States before crossing into Canada.
Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed less than 100 milliseconds 81% of the time, but only 20% of changes to a previous item can be completed in less than this time. If 30% of transactions are changes, what is the probability that a transaction can be completed in less than 100 milliseconds? Round your answer to two decimal places (e.g. 98.76).
Answer:
There is a 62.7% probability that a transaction can be completed in less than 100 milliseconds.
Step-by-step explanation:
This a probability problem.
We have the following probabilities:
-70% probability that a transaction is an addition of an item.
-30% probability that a transaction is a change to an item
-81% probability that an addition can be completed in less than 100 milliseconds
-20% probability that change can be completed in less than 100 milliseconds
The probability P that a transaction can be completed in less than 100 milliseconds is:
[tex]P = P_{1} + P_{2}[/tex]
In which [tex]P_{1}[/tex] is the probability that the transaction is an addition and it takes less than 100 milliseconds. So
[tex]P_{1} = 0.7*0.81 = 0.567[/tex]
[tex]P_{2}[/tex] is the probability that the transaction is a change and it takes less than 100 milliseconds. So
[tex]P_{2} = 0.2*0.3 = 0.06[/tex]
So
[tex]P = P_{1} + P_{2} = 0.567 + 0.06 = 0.627[/tex]
There is a 62.7% probability that a transaction can be completed in less than 100 milliseconds.
Problem 4.28: People with type O-negative blood are
universaldonors. That is, any patient can receive a transfusion
ofO-negative blood. Only 7% of the American population
haveO-negative blood. If 10 people appear at random to give blood,
whatis the probability that at least 1 of them is a
universaldonor?
Answer: 0.516
Step-by-step explanation:
Binomial probability distribution formula to find the probability of getting success in x trial:-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where n is the number of trials , p is the probability of getting success in each trial.
Given : People with type O-negative blood are universal donors.
The proportion of the American population have O-negative blood =0.07
For n=10, the probability that at least 1 of them is a O-negative blood :-
[tex]P(x\geq1)=1-P(x=0)\\\\=1-[^{10}C_0(0.07)^0(1-0.07)^{10}]\\\\=1-[(1)(1)(0.93)^{10}]\ \ \ [\text{ Since}^nC_0=1]\\\\=1-0.483982307179\approx1-0.4840=0.516[/tex]
Hence, the probability that at least 1 of them is a universal donor = 0.516
The height, h, of a ball that is tossed into the air is a function of the time, t, it is in the air. The height in feet for t seconds is given by the function h(t)=−16t2+96t
What is the domain of the function?
Question 8 options:
a)
[0, [infinity] )
b)
(-[infinity], [infinity])
c)
(0, [infinity])
d)
(0, 5)
e)
none
Answer:
e) none
Step-by-step explanation:
The first thing that we need to realise is that this question is set to a practical example of a ball being tossed into the air for a period of time. Thus, we know that:
a) time cannot be negative
b) a ball cannot travel a negative distance in height from where it was tossed it into the air (given that this starting position is also the ground)
In the context of what the problem asks us to achieve, which is to find the domain, we now know that:
a) t ≥ 0
b) the function h(t) is only practical when h(t) ≥ 0
In other words, we know that the domain starts at t = 0, however we need to find when the ball drops back onto the ground. We can do this by solving h(t) = 0:
h(t) = -16t^(2) + 96t
0 = -16t^(2) + 96t
0 = -16t(t - 6) (Factorise -16t^(2) + 96t)
So, now we get:
-16t = 0, therefor t = 0
or
t - 6 = 0, therefor t = 6
Thus, the ball is on the ground at t = 0 seconds and t = 6 seconds; it is between these two values of t that the domain exists and that the problem is practical.
Now, looking at the multiple choice options, it seems as though none of them are correct, therefor the answer would be e) none.
The other way to work through this question (particularly if it is just multiple choice) is that, after realising that the ball is tossed into the air at t = 0 seconds and then drops at some point later, we could already discount a), b) and c) as answers since they include infinity in the domain, which is not practical for this problem.
Then, we could substitute t = 5 into the equation to get:
h(t) = -16(5)^2 + 96*5
h(t) = -16*5*5 + 96*5
h(t) = -80*5 + 96*5
Since we need h(t) to be 0 at the end value of the domain, this is not the correct answer. Thus, the answer is e) none.
Compute the first four terms of the sequence (-2)*k k>=o
Answer:
The fist four terms are 0, -1, -4, -6.
Step-by-step explanation:
Since [tex] k\geq 0 [/tex], the fist four terms of the sequence correspond to those in wich [tex] k [/tex] takes the values 0, 1, 2 and 3, that is
For [tex] k=0 [/tex]: [tex] (-2)*k = (-2)*0 = 0 [/tex]For [tex] k=1 [/tex]: [tex] (-2)*k = (-2)*1 = -2 [/tex]For [tex] k=2 [/tex]: [tex] (-2)*k = (-2)*2 = -4 [/tex]For [tex] k=3 [/tex]: [tex] (-2)*k = (-2)*3 = -6 [/tex]A local fraternity is conducting a raffle where 50 tickets areto be sold--one per customer. There are three prizes to beawarded. If the four organizers of the raffle each buy oneticket, what is the probability that the four organizers
a) win all of the prizes?
b) win exactly two of the prizes?
c) win exactly one of the prizes?
d) win none of the prizes?
The answers:
a) (4) / (19600)
b) (276) / (19600)
c) (4140) / (19600)
d) (15180) / 19600)
Answer:
The answers are the same you stated.
The calculations are in the step-by-step explanation
Step-by-step explanation:
There are 3 prizes to be distributed among 50 tickets. The order these prizes are distributed does not matter. So the total number of prizes is a combination of 3 from 50.
The formula for a combination of n from m is:
[tex]C(m,n) = \frac{m!}{n!(m-n)!}[/tex]
So, the total number of prizes is:
[tex]T = C(50,3) = \frac{50!}{3!(50-3)!} = \frac{50*49*48*47!}{3!*47!} = 19600[/tex]
what is the probability that the four organizers
a) win all of the prizes?
The number of ways that the four organizers can will all of the prizes is a combination of 3 from 4.
[tex]C(4,3) = \frac{4!}{3!1!} = 4[/tex]
The probability that the win all of the prizes is the number of ways that they can win all the prizes divided by the total numbers of ways that the prizes can be distributed.
[tex]P = \frac{4}{19600}[/tex]
b) win exactly two of the prizes?
The total outcomes(total number of ways that the prizes can be distributed) is 19600.
For them to win exactly two of the prizes, we have a combination of 2 from 4(two organizers win prizes) multiplied by a combination of one from 46(one non-organizers wins a prize), so:
[tex]C(4,2)*C(46,1) = \frac{4!}{2!2!}*\frac{46!}{1! 45!} = 6*46 = 276[/tex]
The probability that they win exactly two of the prizes is
[tex]P = \frac{276}{19600}[/tex]
c) win exactly one of the prizes?
The total outcomes(total number of ways that the prizes can be distributed) is 19600.
For them to win exactly one of the prizes, we have a combination of 1 from 4(one organizer wins a prize) multiplied by a combination of two from 46(two non-organizers win prizes), so:
[tex]C(4,1)*C(46,2) = \frac{4!}{1!3!}*\frac{46!}{2! 44!} = 4*1035 = 4150[/tex]
The probability that they win exactly one prize is
[tex]P = \frac{4150}{19600}[/tex]
d) win none of the prizes?
The total outcomes(total number of ways that the prizes can be distributed) is 19600.
For them to win none of the prizes, we have a combination of 3 from 46(3 prizes distributed among 46 non-organizers). So:
[tex]C(46,3) = \frac{46!}{43!3!} = 15180[/tex]
The probability that they don't win any prize is:
[tex]P = \frac{15180}{19600}[/tex]
The probabilities of winning all of the prizes, winning exactly two of the prizes, winning exactly one of the prizes, and winning none of the prizes are [tex]\rm \dfrac{4}{19600},\dfrac{276}{19600},\dfrac{4140}{19600}, \ and \ \dfrac{15180}{19600}[/tex] respectively.
What is probability?Probability means possibility. It deals with the occurrence of a random event. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
Given
A local fraternity is conducting a raffle where 50 tickets are to be sold one per customer.
There are three prizes to be awarded.
If the four organizers of the raffle each buy one ticket.
The total outcomes will be
[tex]\rm ^{50}C_3 = \dfrac{50!}{47! *3!}\\\\\rm ^{50}C_3 = 19600[/tex]
a) win all of the prizes.
The number of ways the four organizers can win all the prizes is given by
[tex]\rm ^4C_3 = \dfrac{4!}{3! *1!}\\\\\rm ^4C_3 = 4[/tex]
Then the probability of winning all the prizes will be
[tex]\rm Probability = \dfrac{4}{19600}[/tex]
b) win exactly two of the prizes.
The number of ways is 2 from organizers and 1 from non-organizers given by
[tex]\rm ^4C_2 * ^{46}C_1 = \dfrac{4!}{2! *2!}*\dfrac{46!}{45! *1!}\\\\\rm ^4C_2 * ^{46}C_1 = 276[/tex]
Then the probability of 2 from organizers and 1 from non-organizers will be
[tex]\rm Probability = \dfrac{276}{19600}[/tex]
c) win exactly one of the prizes.
The number of ways is 1 from organizers and 2 from non-organizers given by
[tex]\rm ^4C_1 * ^{46}C_2 = \dfrac{4!}{1! *3!}*\dfrac{46!}{44! *2!}\\\\\rm ^4C_1 * ^{46}C_2 = 4140[/tex]
Then the probability of 1 from organizers and 2 from non-organizers will be
[tex]\rm Probability = \dfrac{4140}{19600}[/tex]
d) win none of the prizes.
The number of ways is 3 from non-organizers given by
[tex]\rm ^{46}C_3 = \dfrac{46!}{43! *3!}\\\\\rm ^{46}C_3 = 15180[/tex]
Then the probability of 3 from non-organizers will be
[tex]\rm Probability = \dfrac{15180}{19600}[/tex]
Thus, The probabilities of winning all of the prizes, winning exactly two of the prizes, winning exactly one of the prizes, and winning none of the prizes are [tex]\rm \dfrac{4}{19600},\dfrac{276}{19600},\dfrac{4140}{19600}, \ and \ \dfrac{15180}{19600}[/tex] respectively.
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In a biathalon race you first ride a bicycle at an average speed of 21.8 mi/h for 16.5 miles, then you must run for another 5.5 miles. With what average speed, in miles per hour, must you run if your average speed for the entire race is to be 13.8 mi/h?
The average speed for this additional distance is 6.57 mi/h
The given parameters include;
the average speed on bicycle = 21.8 mi/hthe distance traveled on bicycle = 16.5 milesadditional distance to be covered = 5.5 mileslet the average speed for this additional distance = vthe average speed for the entire race = 13.8 mi/hTo find:
the average speed for this additional distance, vThe average speed formula is given as;
[tex]average\ speed = \frac{total \ distance }{total \ time} \\\\average\ speed = \frac{16.5 \ miles \ +\ 5.5 \ miles}{\frac{16.5 \ mi}{21.8 \ mi/h}\ \ +\ \ \frac{5.5 \ mi}{v} } \\\\13.8 \ mi/h = \frac{(16.5 \ miles \ +\ 5.5 \ miles)}{(\frac{16.5 \ mi}{21.8 \ mi/h}\ \ +\ \ \frac{5.5 \ mi}{v} ) }\\\\13.8 = \frac{(16.5 \ +\ 5.5 )}{(\frac{16.5 }{21.8 }\ \ +\ \ \frac{5.5 }{v} ) }\\\\13.8 = \frac{22 }{0.757\ \ +\ \ \frac{5.5 }{v} }\\\\13.8 (0.757\ \ +\ \ \frac{5.5 }{v} ) = 22\\\\[/tex]
[tex]10.446 + \frac{75.9}{v} = 22\\\\\frac{75.9}{v} = 22-10.446\\\\\frac{75.9}{v} = 11.554\\\\v = \frac{75.9}{11.554} \\\\v = 6.57 \ mi/h[/tex]
Thus, the average speed for this additional distance is 6.57 mi/h
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The runner must run at an average speed of 6.57 mi/h to achieve the overall average speed of 13.8 mi/h for the entire biathlon race.
Explanation:To solve for the average running speed needed to achieve an overall average speed of 13.8 mi/h in the biathlon, we will use the concept of weighted averages. The total time for both cycling and running can be expressed in terms of the distances traveled and the speeds for each segment.
First, we calculate the time taken to complete the cycling portion, which is 16.5 miles at an average speed of 21.8 mi/h. The formula for time is distance divided by speed:
Next, we need to find the running speed. Let's call the average running speed 'R'. Using the same formula for time:
The overall average speed for the entire race is the total distance divided by the total time. The total distance is 16.5 miles + 5.5 miles = 22 miles. Let 'T' represent the total time for the race.
Average speed for the entire race = Total distance / Total time = 22 miles / T
Given that the overall average speed must be 13.8 mi/h, we set up the equation:
13.8 mi/h = 22 miles / T
Now we can express 'T' as the sum of the cycling time and running time:
T = Time (cycling) + Time (running) = 0.7565 hours + (5.5 miles / R)
Substitute this into our average speed equation:
13.8 mi/h = 22 miles / (0.7565 hours + (5.5 miles / R))
Now, we solve for 'R', which represents the average running speed. We cross multiply and isolate 'R' to obtain:
(13.8 mi/h) (0.7565 hours + (5.5 miles / R)) = 22 miles
This simplifies to:
10.4469 hours + 75.9 miles / R = 22 miles
Subtract 10.4469 hours from both sides:
75.9 miles / R = 11.5531 miles
Finally, divide 75.9 by 11.5531 to find the average running speed 'R':
R = 6.57 mi/h
The runner must run at an average speed of 6.57 mi/h to have an overall average speed of 13.8 mi/h for the biathlon.
At the grocery store, Hosea has narrowed down his selections to 7 vegetables, 6 fruits, 5 cheeses, and 6 whole grain breads. He wants to use the Express Lane, so he can only buy 15 items. In how many ways can he choose which 15 items to buy if he wants all 5 cheeses?
Answer: 92378
Step-by-step explanation:
Given : At the grocery store, Hosea has narrowed down his selections to 7 vegetables, 6 fruits, 5 cheeses, and 6 whole grain breads.
Total items : 7+6+5+6=24
If he wants all cheeses , then the remaining items needed to be select = 15-5=10
Total items left from which he will select = 24-5=19
No. of combinations of r things out of n : [tex]C(n;r)=\dfrac{n!}{r!(n-r)!}[/tex]
The combination of 10 things selecting from 19 things given by :-
[tex]C(5;5)\timesC(19;10)=\dfrac{19!}{10!(19-10)!}\\\\=(1)\times\dfrac{19\times18\times17\times16\times15\times14\times13\times12\times11\times10!}{10!9!}\ \because ^nC_n=1\\\\=92378[/tex]
Hence, the number of ways he can choose 15 items to buy if he wants all 5 cheeses =92378
Final answer:
If Hosea wants to buy all 5 cheeses, there are 19 items he can choose from the remaining categories. Using the combination formula, the number of ways to choose 10 items from a set of 19 is calculated to be 92378.
Explanation:
If Hosea wants to buy all 5 cheeses, he needs to choose 10 more items from the remaining categories (vegetables, fruits, and whole grain breads). There are 7 vegetables, 6 fruits, and 6 whole grain breads, which means he can choose from a total of 7+6+6 = 19 items in those categories. Since Hosea can only choose 10 more items, he needs to calculate the number of ways to choose 10 items from a set of 19. This can be calculated using the combination formula.
The number of ways to choose 'r' items from a set of 'n' items is given by the combination formula: C(n, r) = n! / (r! * (n-r)!). In this case, n = 19 (the number of items to choose from) and r = 10 (the number of items to choose). Evaluating the formula, we get: C(19, 10) = 19! / (10! * (19-10)!).
By simplifying the factorial expressions and performing the calculations, we find that there are 92378 ways for Hosea to choose his 15 items.
A cold tablet contains the following amounts of active ingredients: acetaminophen 325 mg, chlorpheniramine maleate 2 mg, and dextromethorphan hydrobromide 15 mg. How many tablets may be prepared if a manufacturing pharmacist has 1 kg of acetaminophen, 125 g of chlorpheniramine maleate, and unlimited quantities of dextromethorphan hydrobromide?
Answer:
The number of tablets that can be prepared is 3076.
Step-by-step explanation:
The total amount of active ingredients in the tablet is the sum of the amounts provided in the formula:
[tex]325 mg + 2mg+15 mg=342 mg[/tex]
The percentages of each component in the formula are:
Acetaminophen:[tex]\frac{325mg*100}{342mg}=95.03[/tex]%
Chlorpheniramine maleate:[tex]\frac{2mg*100}{342mg} =0.58[/tex]%
Dextromethorphan hydrobromide:[tex]\frac{15mg*100}{342mg}=4.39[/tex]%
If 1 Kg=[tex]10^{6}[/tex] mg of acetaminophen is used, the needed amount of chlorpheniramine maleate would be:
[tex]\frac{10^{6} mg *0.58}{95.03}=6153.85 mg[/tex]
Since there are 125 g = 125000 mg of chlorpheniramine maleate, there is enough of these ingredient to run the available acetaminophen out. Thus, the total amount of active ingredients that can be prepared with 1 kg of acetaminophen is:
[tex]\frac{10^{6}mg*100}{95.03}=1052307.7mg[/tex]
Since each tablet weighs 342 mg, the number of tablets that can be prepared is:
[tex]\frac{1052307.7mg}{342mg}=3076.923[/tex]
Which means that 3076 tablets can be prepared and a there will be a remanent of 0.923*342 mg = 315.69 mg of active ingredients.
Ingredients are used in ratio to prepare a specific product, generally. The number of tablets that can be manufactured for given context is 3076
How to form mathematical expression from the given description?You can represent the unknown amounts by the use of variables. Follow whatever the description is and convert it one by one mathematically. For example if it is asked to increase some item by 4 , then you can add 4 in that item to increase it by 4. If something is for example, doubled, then you can multiply that thing by 2 and so on methods can be used to convert description to mathematical expressions.
We are given that the considered cold tablet consists:
acetaminophen to chlorpheniramine maleate to dextromethorphan hydrobromide as 325 : 2 : 15 in weight.
Since we're given that there is unlimited quantities of dextromethorphan hydrobromid, the first two ingredients will be the one capping(limiting) the number of tablets that can be manufactured.
Suppose that 'n' tablets can be manufactured by the given amount of ingredients, then:
Amount of acetaminophen needed = 325 mg × n ≤ 1 kg (which is available amount)
Amount of chlorpheniramine maleate = 2 mg × n ≤ 125 g
Converting all scales of weight to grams, we get two inequalities:
[tex]0.325 n \leq 1000[/tex][tex]0.002n \leq 125[/tex](since 1 g = 0.001 mg , and 1 kg = 1000 g)
Remember that 'n' is amount of tablets which is going to be a whole number.
Solving the inequalities, we get:
[tex]0.325 n \leq 1000\\\\n \leq \dfrac{1000}{0.325} = 3076.9\\\\n \leq 3076[/tex]
and
[tex]0.002n \leq 125\\\\n \leq \dfrac{125}{0.002} = 62500[/tex]
So, we see that by the given amount of acetaminophen , we can only make 3076 tablets, but we can make 62500 tablets by the second ingredient chlorpheniramine maleate,
since both ingredients are necessary, so after 3076 tablets, first ingredient will exhaust.
Thus, The number of tablets that can be manufactured for given context is 3076
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At lunchtime, Ciaran buys a sandwich.
He can choose white bread or brown bread.
What is the probability that he chooses brown bread?
Answer:
1/2.
Step-by-step explanation:
There are 2 choices and he has to choose 1 , so the answer is 1/2.
Assuming Ciaran has no preference for white or brown bread, and each choice is equally likely, the probability of choosing brown bread is 1/2 or 50%.
The probability that Ciaran chooses brown bread depends on the assumption that he has no preference and that the choices are equally likely. If the only options available to Ciaran are white bread or brown bread, and each choice is equally likely, then the probability of choosing one over the other is 1 out of the total number of options.
In this case, there are 2 options (white or brown), so the probability that Ciaran will choose brown bread is 1/2 or 0.5, which can also be expressed as a 50% chance.
A long-distance telephone company claims that the mean duration of long-distance telephone calls originating in one town was greater than 9.4 minutes, which is the average for the state. Determine the conclusion of the hypothesis test assuming that the results of the sampling don’t lead to rejection of the null hypothesis.
(A) Conclusion: Support the claim that the mean is less than 9.4 minutes.
(B) Conclusion: Support the claim that the mean is greater than 9.4 minutes.
(C) Conclusion: Support the claim that the mean is equal to 9.4 minutes.
(D) Conclusion: Do not support the claim that the mean is greater than 9.4 minutes.
Answer:
(D) Conclusion: Do not support the claim that the mean is greater than 9.4 minutes
Step-by-step explanation:
Let X be the mean duration of long distance telephone class originating in one town.
[tex]H_0: x bar = 9.4\\H_a: x bar >9.4[/tex]
(one tailed test)
The conclusion of the hypothesis test assuming that the results of the sampling don’t lead to rejection of the null hypothesis.
This means that there is no statistical evidence to support the alternate claim.
Hence option D is right.
(D) Conclusion: Do not support the claim that the mean is greater than 9.4 minutes
Solve the initial value problem: dydx+5y=7 y(0)=0
Answer:
Given differential equation,
[tex]\frac{dy}{dx}+5y=7[/tex]
[tex]\frac{dy}{dx}=7-5y[/tex]
[tex]\implies \frac{dy}{7-5y}=dx[/tex]
Taking integration both sides,
[tex]\int \frac{dy}{7-5y}=\int dx[/tex]
Put 7 - 5y = u ⇒ -5 dy = du ⇒ dy = -du/5,
[tex]-\frac{1}{5} \int \frac{du}{u} = \log x + C[/tex]
[tex]-\frac{1}{5} \log u = \log x + C[/tex]
[tex]-\frac{1}{5}\log(7-5y) = \log x + C---(1)[/tex]
Here, x = 0, y = 0
[tex]\implies -\frac{1}{5} \log 7= C[/tex]
Hence, from equation (1),
[tex]-\frac{1}{5}\log(7-5y)=\log x -\frac{1}{5}log 7[/tex]
[tex]\log(7-5y)=\log (\frac{x}{7^\frac{1}{5}})[/tex]
[tex]7-5y=\frac{x}{7^\frac{1}{5}}[/tex]
[tex]7-\frac{x}{7^\frac{1}{5}}=5y[/tex]
[tex]\implies y=\frac{1}{5}(7-\frac{x}{7^\frac{1}{5}})[/tex]
A bacteria culture starts with 200 bacteria and grows at a rate proportional to its size. After 6 hours there will be 1200 bacteria (1) Express the population after I hours as a function of t. population: p(tepe (1.066-21) (unction of t) (b) What will be the population after 7 hours? 348125.2 (c) How long will it take for the population to reach 1750 ? Note: You can earn partial credit on this problem.
Answer:
We are given that the rate of change is proportional to its size S
So, [tex]\frac{dS}{dt} \propto S[/tex]
[tex]\frac{dS}{dt} = kS[/tex]
[tex]\frac{dS}{S} = kdt[/tex]
Integrating both sides
[tex]\log(S)= kt + log c[/tex]
[tex]\frac{S}{S_0}=e^{kt}[/tex]
[tex]S=S_0 e^{kt}[/tex]
S is the population after t hours
[tex]S_0[/tex] is the initial population
Now we are given that After 6 hours there will be 1200 bacteria
[tex]1200=200 e^{6k}[/tex]
[tex]6=e^{6k}[/tex]
[tex]6^{\frac{1}{6}=e^{k}[/tex]
So, [tex]S=200 \times 6^{\frac{t}{6}[/tex]
a)Now the population after t hours as a function of t; [tex]S=200 \times 6^{\frac{t}{6}[/tex]
b) What will be the population after 7 hours?
Substitute t = 7 hours
A bacteria culture starts with 200 bacteria
[tex]S=200 \times 6^{\frac{7}{6}}[/tex]
[tex]S=1617.607[/tex]
c) How long will it take for the population to reach 1750 ?
[tex]1750=200 \times 6^{\frac{t}{6}[/tex]
[tex]\frac{1750}{200} =6^{\frac{t}{6}[/tex]
[tex]8.75 =6^{\frac{t}{6}[/tex]
[tex]t=7.26[/tex]
So, it will take 7.2 hours for the population to reach 1750
Final answer:
To determine the population growth function in terms of time and find the population after a specific duration, use the exponential growth formula N(t) = N0 x 2^t. Calculate the growth rate using given data points like the initial and final population. Finally, substitute the desired time into the population function to find the population at that specific time.
Explanation:
The population of bacteria after t hours can be represented by the formula N(t) = N0 x 2t.
(a) To express the population after t hours as a function of t, you can use the given data points to find the growth rate. For the provided data, the growth rate is calculated as r = log2(1200/200) / 6 = 0.1333 per hour.
(b) To find the population after 7 hours, substitute t=7 into the function: N(7) = 200 x 27x0.1333 = 3481.49.
c) To find out how long it will take for the population to reach 1750, we get
p(t)=1750 and solving it
t ≈7.846
Suppose that a Petri dish initially contains 2000 bacteria cells. An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. Let P(t) be the number of bacteria cells present t hours after the antibiotic is introduced. (a) (8 points) Suppose that P(t) is a linear function. Find a formula for P(t) (b) (8 points) Suppose that P(t) is an exponential function. Find a formula for P(t)
Answer:
a)The linear function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000 - 100*t[/tex]
b)The exponential function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000e^{-0.055t}[/tex]
Step-by-step explanation:
(a) Suppose that P(t) is a linear function. Find a formula for P(t):
[tex]P(t)[/tex] can be modeled by a linear function in the following format.
[tex]P(t) = P_{0} - r*t[/tex], in which [tex]P_{0}[/tex] is the initial number of bacteria cells in the dish, t is the time and r is the rate that the number decreases.
Since the dish initially contains 2000 bacteria cells, [tex]P_{0} = 2000[/tex]
We have
[tex]P(t) = 2000 - r*t[/tex]
An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. So [tex]P(4) = 1600[/tex]. With this information, we can find the value of r.
[tex]P(t) = 2000 - r*t[/tex]
[tex]1600 = 2000 - r*(4)[/tex]
[tex]4r = 400[/tex]
[tex]r = \frac{400}{4}[/tex]
[tex]r = 100[/tex]
So, the linear function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000 - 100*t[/tex]
b) Suppose that P(t) is an exponential function. Find a formula for P(t)
[tex]P(t)[/tex] can also be modeled by an exponential function in the following format:
[tex]P(t) = P_{0}e^{rt}[/tex]
The values mean the same as in a). We use the fact that [tex]P(4) = 1600[/tex] to find r.
[tex]P(t) = 2000e^{rt}[/tex]
[tex]1600 = 2000e^{4r}[/tex]
[tex]e^{4r} = \frac{1600}{2000}[/tex]
[tex]e^{4r} = 0.8[/tex]
[tex]ln e^{4r} = ln 0.8[/tex]
[tex]4r = -0.22[/tex]
[tex]r = \frac{-0.22}{4}[/tex]
[tex]r = -0.055[/tex]
So, the exponential function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000e^{-0.055t}[/tex]
Rosie washes clothes for two families the first family pays her $500 more per month than the second. Her total earning per month is $3200. How much does she earn from each families?
Answer:
Family 1: $1.850Family 2: $1.350Step-by-step explanation:
We know that the total monthly payment is 3200, so if we call Pa (family A`s payment) and Pb (Family B's payment) the payments:
Pa+Pb=3200also, Pa = Pb+500So if we replace Pa in the first ecuation:
Pb+500+Pb=32002Pb= 3200-500Pb=2700/2= $1.350then Pa+Pb=3200 => Pa= 3200-1350= $1.850
Good Luck!
suppose you deposit $1000 in an account paying 4.6% annual interest compounded continuously. How long will it take for the money to double?
Answer: About 16 years
Step-by-step explanation:
The formula to find the compound amount if compounded continuously is given by :-
[tex]A=Pe^{rt}[/tex], where P is Principal amount, r is the rate of interest ( in decimal) and t is time ( in years).
Given : P= $1000 ; r= 4.6%=0.046
let t be the time it will take to double the amount, the we have
[tex]2(1000)=(1000)e^{0.046\times t}[/tex]
Dividing 1000 both sides, we get
[tex]2=e^{0.046 t}[/tex]
Taking natural log on each side, we get
[tex]\ln2=\ln(0.046\times t)\\\\\Rightarrow\ 0.6931=0.046t\\\\\Rightarrow\ t=\dfrac{0.6931}{0.046}=15.0673913043\approx16\text{ years}[/tex]
Hence, it will take about 16 years to double the amount.
Plot another sin function of 20% higher frequency over the same range.
Step-by-step explanation:
The frequency of sine function is given by the number of periods in a given range. For example:
Frequency for [tex]sin(x)[/tex] is 1 in the interval [tex][0,2\pi][/tex].This means that, if we want another sine function with frequency 20% higher, we need that function to have a frequency of 1.2 in the interval [tex][0,2\pi][/tex].
To be easier to see we will consider interval [tex][0,10\pi][/tex] instead of [tex][0,2\pi][/tex]. In this interval [tex]sin(x)[/tex] has 5 periods, therefore our new sine function should have 6 periods.
Finally, as we can see in the graph, the function [tex]sin(\frac{6}{5}x )[/tex] (in blue) has a frequency 20% higher than [tex]sin(x)[/tex] (in red). This can be easily seen counting the number of periods between 0 and [tex]10\pi[/tex] for both functions. 5 for [tex]sin(x)[/tex] and 6 for [tex]sin(\frac{6}{5} x)[/tex].
You are looking down at a map. A vector u with |u| = 6 points north and a vector v with |v| = 5 points northeast. The crossproduct u×v points:
(A) south
(B) northwest
(C) up
(D) down
(E) The magnitude |u×v| =
Answer: Hi!
First, UxV = sin(a)*IUI*IVI
where a is the angle between U and V, in this case 45°.
First, the cross product of UxV points:
Here you can use the right hand method,
Put your hand flat, so the point of your fingers point in the same direction that the first vector, in this case U, so your fingers will point to the north.
Now roll your fingers in the direction of the second vector, so here you will roll your fingers in the northeast direction. Now you will see that your thumb is pointing down, so the cross product of UxV points down.
And the magnitude is 6*5*sin(45) = 21.213
The cross-product u×v points: Option D: Down, and its magnitude |u×v| evaluates to [tex]|u \times v| = 15\sqrt{2}[/tex]
How to find the cross product of two vectors?Suppose that two vectors in consideration are u and v, then their cross product is evaluated as:
[tex]u \times v = |u|.|v|.sin(\theta)\hat{n}[/tex]
where [tex]\hat{n}[/tex] is the normal unit vector whose direction is decided by right hand thumb rule, and theta is the angle between u and v vector.
The two bars around a vector represents the magnitude of that vector.
Cross product returns the result as a vector itself.
For this case, we have:
|u| = 6 points, its direction is in north|v| = 5 points, its direction is in northeastThus, as north and northeast have 45 degrees in between them, therefore, we get:
[tex]u \times v = 6\times 5 \times sin(45^\circ) \hat{n} = 15\sqrt{2} \: \hat{n}[/tex]
Directing index finger of right hand to north direction, and middle to northeast makes the thumb go down, therefore, the direction of normal vector (and therefore direction of the resultant cross product vector too) is downside of this whole north south east west plane.
The magnitude of cross product is [tex]|u \times v| = 15\sqrt{2}[/tex]
Thus, the cross-product u×v points: Option D: Down, and its magnitude |u×v| evaluates to [tex]|u \times v| = 15\sqrt{2}[/tex]
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Clearly write what it means for a set to be closed under an operation
Answer:
A set S is closed under an operation * (we're denoting the operation as asterisk) IF for any two elements a,b in S, the result a*b is also in S.
Step-by-step explanation:
A set being closed under an operation means that whenever you operate elements from the set, the result you get out of it is ALWAYS inside the set. For example, think of the set Z of integer numbers and the operation + (usual addition). If we add ANY two integers, we're going to get another integer. Or said in terms of sets, for any two numbers a,b in Z, a+b is also in Z.
On the other side, not being closed under an operation means you do NOT ALWAYS get results inside the same set. Think of the set of natural numbers N, and the operation - (usual difference). If we do the operation 5-12, we get -7 which is NOT in the set of natural numbers. So N is not closed under subtraction.
Use the roster method to write each of the given sets. For some exercises you may need to consult a reference, such as the Internet or an encyclopedia. (Enter EMPTY for the empty set.)
The set of natural numbers x that satisfy x + 2 = 1
There are no natural numbers that satisfy the equation x + 2 = 1. Therefore, using the roster method, the set is empty.
Explanation:The question requires us to use the roster method to write the set of natural numbers x that satisfy the equation x + 2 = 1. Natural numbers, by definition, are counting numbers starting from 1. They are non-negative and do not include zero. So, if we try to find a natural number x that satisfies the equation x + 2 = 1, we see that x would need to be -1 (since -1 + 2 equals 1). However, -1 is not a natural number. Therefore, there are no natural numbers that satisfy the equation, so the set is empty.
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The set of natural numbers x that satisfy x + 2 = 1. The correct answer is EMPTY.
To solve the equation x + 2 = 1 for natural numbers x we would first try to isolate x subtracting 2 from both sides of the equation:
x + 2 - 2 = 1 - 2
x = -1
However, natural numbers are defined as the set of positive integers, starting from 1 and increasing indefinitely.
Since the solution to the equation x = -1 is not a positive integer, it does not belong to the set of natural numbers. Therefore, there is no natural number x that satisfies the equation x + 2 = 1
Since there are no elements that satisfy the condition, the set is empty. Hence, the correct representation of the set using the roster method is EMPTY.
Jamie just paid off a loan he took out six months ago at 12% simple annual interest. He paid $3,816.00, which was the sum of the principal and the simple interest accrued over the length of the loan. What was the amount of principal he borrowed? Give answer in dollars rounded to the nearest cent. Do NOT enter "$" sign in answer
Answer:
The amount of principal Jamie borrowed was $3600.
Step-by-step explanation:
Let the principal of loan is = Po
[tex]P=Po\times0.12\times0.5[/tex]
Here r = 12% or 0.12
t = 6 months or 0.5 years
Now as given : Po+Interest=3816
[tex]Po+0.06Po=3816[/tex]
[tex]=>1.06Po=3816[/tex]
So, Po=3600
Hence, the amount of principal Jamie borrowed was $3600.
8/3(z+11)=y solve for z.
Answer:
z = (3/8)y - 11
Step-by-step explanation:
Undo what has been done to z, in reverse order. Here, 11 is added and the sum is multiplied by 8/3.
To undo the multiplication, multiply the equation by the inverse, 3/8:
(3/8)(8/3)(z+11) = (3/8)y
z +11 = (3/8)y . . . . . . . . . simplify
To undo the addition of 11, add the opposite of 11 to the equation.
z +11 -11 = (3/8)y -11
z = (3/8)y -11 . . . . . . . . . simplify
Rewrite the following statements in if then form a) Catching the 8:05 bus is a sufficient condition for my being on time for work b) Being divisible by 3 is a necessary condition for this number to be divisible by 9 o) The Cubs will win the pennant only if they win tomorrow's game.
Answer:
a. If I Catch the 8:05 bus, then I will arrive on time for work.
b. If an integer is divisible by 9, then that integer is divisible by 3.
C. If the Cubs win tomorrow's game, then they win the pennant and if the Cubs have won the pennant it is because they will have won the game tomorrow.
Step-by-step explanation:
Final answer:
To rewrite the statements in 'If...Then' form: 1. If you catch the 8:05 bus, then you will be on time for work. 2. If a number is divisible by 9, then it is divisible by 3. 3. If the Cubs win the pennant, then they have won tomorrow's game.
Explanation:
Rewriting Statements in 'If...Then' Form
To rephrase the given statements in 'If...Then' form while identifying the sufficient and necessary conditions, we'll look at each statement individually:
Catching the 8:05 bus is a sufficient condition for being on time for work. This can be rewritten as: If you catch the 8:05 bus, then you will be on time for work.
Being divisible by 3 is a necessary condition for a number to be divisible by 9. The 'If...Then' form is: If a number is divisible by 9, then it is divisible by 3.
The Cubs will win the pennant only if they win tomorrow's game. In the 'If...Then' form: If the Cubs win the pennant, then they have won tomorrow's game.
Each of these rephrased statements now clearly shows the reliance of the consequent (the outcome) on the antecedent (the condition).
Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.
A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $4,000 or $8,000. If the partnership raised $348,000, then how many investors contributed $4,000 and how many contributed $8,000?
x = $4,000 investors
y =
$8,000 investors
Solve the system by row-reducing the corresponding augmented matrix. (Enter your answers as a comma-separated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)
leftbrace2.gif
2x + y = 17
x + y = 13
the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.
A jar contains 80 nickels and dimes worth $6.80. How many of each kind of coin are in the jar?
x = nickels
y = dimes
Answer:
1) There were 33 $4,000 investors and 27 $8,000 investors.
2) The solution in x = 4, y = 9.
3) There were 24 nickels and 56 dimes.
Step-by-step explanation:
1) A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $4,000 or $8,000. If the partnership raised $348,000, then how many investors contributed $4,000 and how many contributed $8,000?
I am going to say that:
x is the number of investors that contributed 4,000.
y is the number of investors that contributed 8,000.
Building the system:
There are 60 investors. So:
[tex]x + y = 60[/tex]
In all, the partnership raised $348,000. So:
[tex]4000x + 8000y = 348000[/tex]
I am going to simplify by 4000. So:
[tex]x + 2y = 87[/tex]
Solving the system:
The elimination method is a method in which we can transform the system such that one variable can be canceled by addition. So:
[tex]1)x + y = 60[/tex]
[tex]2)x + 2y = 87[/tex]
I am going to multiply 1) by -1. So we have
[tex]1)-x - y = -60[/tex]
[tex]2)x + 2y = 87[/tex]
By addition, the x are going to cancel each other
[tex]-x + x - y + 2y = -60 + 87[/tex]
[tex]y = 27[/tex]
For x:
[tex]x + y = 60[/tex]
[tex]x = 60-y = 60-27 = 33[/tex]
There were 33 $4,000 investors and 27 $8,000 investors.
2) Solve the system by row-reducing the corresponding augmented matrix.
[tex]2x + y = 17[/tex]
[tex]x + y = 13[/tex]
This system has the following augmented matrix:
[tex]\left[\begin{array}{ccc}2&1&17\\1&1&13\end{array}\right][/tex]
To help the row reducing, i am going to swap the first with the second line:
[tex]L1 <-> L2[/tex]
So we have:
[tex]\left[\begin{array}{ccc}1&1&13\\2&1&17\end{array}\right][/tex]
Now, reducing the first column.
[tex]L2 = L2 - 2L1[/tex]
So we have:
[tex]\left[\begin{array}{ccc}1&1&13\\0&-1&-9\end{array}\right][/tex]
Now we do:
[tex]L2 = -L2[/tex]
And the matrix is:
[tex]\left[\begin{array}{ccc}1&1&13\\0&1&9\end{array}\right][/tex]
Now to reduce the second column, we do:
[tex]L1 = L1 - L2[/tex]
[tex]\left[\begin{array}{ccc}1&0&4\\0&1&9\end{array}\right][/tex].
So the solution is:
x = 4, y = 9.
3) A jar contains 80 nickels and dimes worth $6.80. How many of each kind of coin are in the jar?
I am going to say that x is the number of nickels and y is the number of dimes.
Each nickel is worth 5 cents and each dime is worth 10 cents.
Building the system:
There are 80 coins in all:
[tex]x + y = 80[/tex]
They are worth $6.80. So:
[tex]0.05x + 0.10y = 6.80[/tex]
Solving the system:
[tex]1)x + y = 80[/tex]
[tex]2)0.05x + 0.10y = 6.80[/tex]
I am going to divide 1) by -10, so we can cancel y. So:
[tex]1)-0.10x - 0.10y = -8[/tex]
[tex]2)0.05x + 0.10y = 6.80[/tex]
Adding:
[tex]-0.10x + 0.05x - 0.10y + 0.10y = -8 + 6.80[/tex]
[tex]-0.05x = -1.2[/tex] *(-100)
[tex]5x = 120[/tex]
[tex]x = \frac{120}{5}[/tex]
[tex]x = 24[/tex]
Also
[tex]x + y = 80[/tex]
[tex]y = 80-x = 80-24 = 56[/tex]
There were 24 nickels and 56 dimes.
Which expressions are equal to 11x10^3? Select all that apply.
A. 11x100
B. 11,000
C. 11x10x10x10
D. 33,000
E. 11x11x11
The required simplified expression for 11 x 10³ is 11,000. Option B is correct.
Given that,
A simplified expression form of the expression 11 x 10³, is to be determined.
The process in mathematics to operate and interpret the function or expression to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
= 11 x 10³
Since 10³ = 10 x 10 x 10 = 1000
= 11 x 1000
= 11,000
Thus, the required simplified expression for 11 x 10³ is 11,000. Option B is correct.
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