Answer with Step-by-step explanation:
Let A= {a, b,c,d} and B= {1, 2, 3, 4, 5, 6}
Define f:A→B
where f = {(a, 1), (b, 3), (c, 5), (d, 2)}
a. What is the co-domain of f
The co-domain or target set of a function is the set into which all of the output of the function is constrained to fall.
Here, all the values are constrained to fall on the set B
Hence, co-domain={1, 2, 3, 4, 5, 6}
b. What is the range of f?
The set of all output values of a function is called range.
Here, on putting the values of set A we get the output values as:1,2,3 and 5
Hence, Range={1,2,3,5}
c. Is f 1-1 function Explain.
A function for which every element of the range of the function corresponds to exactly one element of the domain.
Yes, f is 1-1
(Since, every element of range i.e. 1,2,3 and 5 corresponds to only one element of set A)
d. Can there exist a bijection between A and B? Explain.
No, there cannot exist bijection from A to B
because if a bijection exist between two sets then there cardinalities are same but A and B have different cardinalities
A has cardinality 4 and B has cardinality 6
The people at a party tried to form teams with the same number of people on each team, but when they tried to split up into teams of 2, 3, 5, or 7, exactly one person was left without a team. What is the smallest amoutn of people who could have been at the party?
Answer:
211 people
Step-by-step explanation:
1. This is a least common multiple question, otherwise known as an LCM question. We know this because if the one extra person had not shown up to the party, all groups would have been formed evenly. This means the amount of people at the party is 1 more than a number 2, 3, 5, and 7 can go into.
2. We also know this is an LCM question because we are being asked for the smallest amount of people who could have possibly attended the party.
3. From this we know the amount of people at the party must be an odd number. If the number were even, there would have been no left over when groups of 2 were formed.
4. The amount of people at the party must end in a 1. This is because all multiples of 5 always end in a 0 or 5. Because there is on extra person, we must add 1 to all multiples of 5 we check. However, 5 + 1 is 6. This is a problem because 6 is an even number, and as we already established, the amount of people at the party must end in an odd number. So, we now know the smallest amount of people at the party will end in a 1.
5. Because we know the largest teams attempted to be formed with 1 left over is teams of 7 people, we only need to check multiples of 7. It is the largest number, so doing this will save us time.
6. Since we know the amount of people at the party must end in a 1, and we are only checking multiples of 7, we only need to check multiples of 7 that end in a 0. This is because any multiple of 7, 7 will go into evenly without a remainder. So we must add 1 to every multiple we check in order to make a remainder of 1. The only number we can add 1 to in order to get 1 is 0, so we only need to check multiples of 7 that end in 0. The only multiples of 7 that end in a 0 are when 7 is multiplied by a ten, ex: 10, 20, 30, 40, 50, ect.
7. Only searching for odd numbers, numbers that end in 1, and multiples of 7 means we only have to check if any possible answer when divided by 3 has a remainder of 1. We only have to check by the number 3 because any number ending in 1 will automatically have a remainder of one for 2 and 5, and because we are using multiple of 7 we don't need to check through 7.
8. Now that we know all our rules, all we need to do is list multiples of 7 that end in 0. Then we will add 1 to them and check to see if they have a remainder of 1 when divided by 3.
Using these rules will narrow our search drastically.
Applicable multiples of 7
(7 × 10) = 70 → 70 + 1 = 71 → 71/3 = 23 R2 → not possible
(7 × 20) = 140 → 140 + 1 = 141 → 141/3 = 47 → not possible
(7 × 30) = 210 → 210 +1 = 211 → 211/3 = 70 R1 → possible
The smallest possible amount of people at the party is 211.
Final answer:
The smallest amount of people who could have been at the party is 210.
Explanation:
To find the smallest amount of people who could have been at the party, we need to find the least common multiple (LCM) of the numbers 2, 3, 5, and 7. The LCM is the smallest number that is divisible by all of the given numbers without leaving a remainder.
Prime factorize each of the given numbers:Therefore, the smallest amount of people who could have been at the party is 210.
A textbook store sold a combined total of 440 physics and sociology textbooks in a week. The number of sociology textbooks sold was 54 less than the number of physics textbooks sold. How many textbooks of each type were sold?
Step-by-step explanation:
If p is the number of physics books and s is the number of sociology books, then:
p + s = 440
s = p - 54
Substituting:
p + (p - 54) = 440
2p - 54 = 440
2p = 494
p = 247
Solving for s:
s = p - 54
s = 247 - 54
s = 193
The store sold 247 physics books and 193 sociology books.
The foreman of a bottling plant has observed that the amount of soda in each \16-ounce" bottle is actually a normally distributed random variable, with a mean of 15.9 ounces and a standard deviation of 0.1 ounce. If a customer buys one bottle, what is the probability that the bottle will contain more than 16 ounces
Answer: 0.1587
Step-by-step explanation:
Given : The foreman of a bottling plant has observed that the amount of soda in each 16-ounce bottle is actually a normally distributed random variable, with
[tex]\mu=15.9\text{ ounces}[/tex]
Standard deviation : [tex]\sigma=0.1\text{ ounce}[/tex]
Let x be the amount of soda in a randomly selected bottle.
Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]
[tex]z=\dfrac{16-15.9}{0.1}=1[/tex]
The probability that the bottle will contain more than 16 ounces using standardized normal distribution table :
[tex]P(x>16)=P(z>1)=1-P(z<1)\\\\=1-0.8413447=0.1586553\approx0.1587[/tex]
Hence, the probability that the bottle will contain more than 16 = 0.1587
Consider a periodic review system. The target inventory level is 1000 units. It is time to review the item, and the on-hand inventory level is 200 units. How many units should be ordered?
a) 800
b) 1000
c) 1200
d) the EOQ amount
e) the safety stock amount
Answer:
The answer is - a) 800
Step-by-step explanation:
In a periodic review system, we calculate the quantity of an item, a company has on hand at specified and fixed interval of time.
Given is : The target inventory level is 1000 units and the on-hand inventory level is 200 units.
So, the quantity will be =[tex]1000-200=800[/tex] units.
The answer is option A.
g 4. Determine which of the following functions are even, which are odd, and which are neither. (a) f(x) = x 3 + 3x (b) f(x) = 4 sin 2x (c) f(x) = x 2 + |x| (d) f(x) = e x (e) f(x) = 1 x (f) f(x) = 1 2 (e x + e −x ) (g) f(x) = x cos x (h) f(x) = 1 2 (e x − e −x ).
Answer:Given below
Step-by-step explanation:
A function is said to be odd if
[tex]F\left ( x\right )=F\left ( -x\right )[/tex]
(a)[tex]F\left ( x\right )=x^3+3x[/tex]
[tex]F\left ( -x\right )=-x^3-3x=-\left ( x^3+3x\right )[/tex]
odd function
(b)[tex]F\left ( x\right )=4sin2x[/tex]
[tex]F\left ( -x\right )=-4sin2x[/tex]
odd function
(c)[tex]F\left ( x\right )=x^2+|x|[/tex]
[tex]F\left ( -x\right )=\left ( -x^2\right )+|-x|=x^2+|x|[/tex]
even function
(d)[tex]F\left ( x\right )=e^x[/tex]
[tex]F\left ( -x\right )=e^{-x}[/tex]
neither odd nor even
(e)[tex]F\left ( x\right )=\frac{1}{x}[/tex]
[tex]F\left ( -x\right )=-\frac{1}{x}[/tex]
odd
(f)[tex]F\left ( x\right )=\frac{1}{2}\left ( e^x+e^{-x}\right )[/tex]
[tex]F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )[/tex]
even function
(g)[tex]F\left ( x\right )=xcosx(h)[/tex]
[tex]F\left ( -x\right )=-xcosx(h)[/tex]
odd function
(h)[tex]F\left ( x\right )=\frac{1}{2}\left ( e^x-e^{-x}\right )[/tex]
[tex]F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )[/tex]
odd function
Q8. the average Ferris wheel rotates at 6.9 miles per hour. What circular distance, in feet dose the average Ferris wheel cover in a 5 minutes ride?
Answer:
303.6 feet
Step-by-step explanation:
Given,
The average Ferris wheel rotates at 6.9 miles per hour.
So, the speed of the wheel = 6.9 miles per hour,
We know that,
Distance = Speed × Time
So, the distance covered by the wheel in 5 minutes ( or 1/12 hours because 1 hour = 60 minutes ) ride = [tex]6.9\times \frac{1}{12}[/tex]
[tex]=\frac{6.9}{12}[/tex]
[tex]=0.575\text{ miles}[/tex]
Since, 1 mile = 5280 feet,
Hence, the distance covered by the wheel in 5 minutes = 0.575 × 528 = 303.6 feet.
The average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.
Explanation:To calculate the circular distance covered by the average Ferris wheel in a 5-minute ride, we need to convert the speed from miles per hour to feet per minute. There are 5,280 feet in a mile and 60 minutes in an hour, so we can convert 6.9 miles per hour to feet per minute using the formula:
6.9 miles/hour x 5,280 feet/mile x 1 hour/60 minutes = 604.8 feet/minute
Now that we know the Ferris wheel covers 604.8 feet in 1 minute, we can calculate the circular distance covered in 5 minutes by multiplying the feet per minute by the number of minutes:
604.8 feet/minute x 5 minutes = 3024 feet
Therefore, the average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.
Assume that women's heights are normally distributed with a mean given by mu equals 62.3 in, and a standard deviation given by sigma equals 2.4 in.(a) If 1 woman is randomly selected, find the probability that her height is less than 63 in.(b) If 47 women are randomly selected, find the probability that they have a mean height less than 63 in.
Answer: a) 0.6141
b) 0.9772
Step-by-step explanation:
Given : Mean : [tex]\mu= 62.3\text{ in}[/tex]
Standard deviation : [tex]\sigma = \text{2.4 in}[/tex]
The formula for z -score :
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
a) Sample size = 1
For x= 63 in. ,
[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{1}}}=0.29[/tex]
The p-value = [tex]P(z<0.29)=[/tex]
[tex]0.6140918\approx0.6141[/tex]
Thus, the probability is approximately = 0.6141
b) Sample size = 47
For x= 63 ,
[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{47}}}\approx2.0[/tex]
The p-value = [tex]P(z<2.0)[/tex]
[tex]=0.9772498\approx0.9772[/tex]
Thus , the probability that they have a mean height less than 63 in =0.9772.
An environmentalist wants to find out the fraction of oil tankers that have spills each month.Step 1 of 2:Suppose a sample of 474tankers is drawn. Of these ships, 318 did not have spills. Using the data, estimate the proportion of oil tankers that had spills. Enter your answer as a fraction or a decimal number rounded to three decimal places.
Answer: The proportion of oil tankers that had spills is [tex]\dfrac{156}{474}[/tex] or 0.329.
Step-by-step explanation:
Since we have given that
Number of tankers is drawn = 474
Number of tankers did not have spills = 318
Number of tankers have spills = 474 - 318 = 156
Proportion of oil tankers that had spills is given by
[tex]\dfrac{Containing\ spill}{Total}=\dfrac{156}{474}=0.329[/tex]
Hence, the proportion of oil tankers that had spills is [tex]\dfrac{156}{474}[/tex] or 0.329.
ASAP PLEASE RESPOND
To win the game, Eitan has to roll a sum of 11 or more using two six-sided number cubes.
Asher has a better probability of winning than Eitan has. Which could be the outcome that Asher needs to win the game? Check all that apply.
rolling a sum of 4
rolling a sum of 9
rolling a sum that is less than 5
rolling a sum that is greater than 5 but less than 7
rolling a sum that is greater than 9 but less than 11
rolling a sum that is greater than 2 but less than 4
There are 36 total possible outcomes.
Rolling a sum of 11 or higher, there are 3 possible rolls, to make a 3/36 = 1/12 probability.
Rolling a sum of 4 there are also 3 possibilities, so the chance would be the same.
Rolling a sum of 9, there are 4 possibilities, which is a better chance.
Rolling a sum less than 5, there is 6 possibilities, which is a better chance.
Rolling greater than 5 but less than 7 means rolling a sum of 6, there are 5 chances, which is a better chance.
Rolling greater than 9 but less than 11, means rolling a 10, there are 3 possibilities, which is the same.
Rolling greater than 2 and less than 4 means rolling a 3, there are 2 possibilities, which is less.
The answers would be:
Rolling a sum of 9,
Rolling a sum less than 5
Rolling greater than 5 but less than 7
Answer:
B, C, D
Step-by-step explanation:
I got it right on Edg
1. A researcher is interested in studying whether or not listening to music while jogging makes people run faster. He thinks that listening to music will make people run faster. Luckily, he knows that, for the population he interested in (runners in Washington, DC), the mean running speed (μ) is 6mph, and the standard error is 2mph. He collects data from a sample of runners that only listen to music, and finds they have a mean running speed (M) of 9mph. No Sample Size givenA. State the hypothesisH0:H1:B. The researcher would like to conduct a One-Sample Z-test. Please calculate the Z-statistic (Z-obtained):
Step-by-step explanation:
1.Assuming the same sample size and considering the same value for the errors ( not taking into consideration the type of music, the volume of the sound and cow familiar the runner is with that type of stimuli, age group, time of the day/ number of days, running conditions like wether and equipment, distance) one can state:
A. Music has no influence over the running speed ( when jogging) in Washington DC
B.When listening to music, people ( in Washington DC) run faster while jogging
The mean running speed is a simple, ponderate or other type of mean ( that takes into consideration the variations of speed at the beginning and by the end of the race?
An experimental psychologist is interested in whether the color of an animal's surroundings affects learning rate. He tests 16 rats in a box with colorful wallpaper. The average rat (of this strain) can learn to run this type of maze in a box without any special coloring in an average of 25 trials, with a variance of 64, and a normal distribution. The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper, is 11.What is the μM ?• A. 8• B. 11• C. 25• D. 64
Answer: C. 25
Step-by-step explanation:
Given : The average rat (of this strain) can learn to run this type of maze in a box without any special coloring : [tex]\mu=25[/tex]
The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper= [tex]M= 11[/tex]
We know that the sampling distribution D is given by :-
[tex]\mu_D=\mu[/tex]
Similarly the mean of the distribution M in the given situation is given by :_
[tex]\mu_M=\mu=25[/tex]
The mean of the distribution M in the given situation is 25. Then the correct option is C.
What is normal a distribution?It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.
An experimental psychologist is interested in whether the color of an animal's surroundings affects the learning rate.
He tests 16 rats in a box with colorful wallpaper.
The average rate (of this strain) can learn to run this type of maze in a box without any special coloring in an average of 25 trials, with a variance of 64, and a normal distribution.
The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper, is 11.
We know that the sampling distribution D is given by
μD = μ
Similarly, the mean of the distribution M in the given situation is given by
μD = μ = 25
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At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
Step-by-step explanation:
I just found the answer and I hope that this helps :)!!
The rate at which the distance between the ships is changing at 4 PM depends on their velocities.
Explanation:To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity. Let's consider ship B as the reference point. Ship A is moving west at 20 knots (which is equivalent to 20 nautical miles per hour), and ship B is moving north at 21 knots. The distance between the ships can be considered as the hypotenuse of a right triangle, with the velocities of the ships representing the triangle's sides.
Using the Pythagorean theorem, we can write the equation: d^2 = x^2 + y^2, where d is the distance between the ships, x is the velocity of ship A, and y is the velocity of ship B. We need to find the rate of change of d with respect to time (dt).
Taking the derivative on both sides of the equation with respect to time, we get: 2d * (dd/dt) = 2x * (dx/dt) + 2y * (dy/dt).
Substituting the given values, x = -20 knots (negative because ship A is moving west), y = 21 knots, and dx/dt = dy/dt = 0 (since ship B is not changing its velocity), we can solve for dd/dt, which represents the rate at which the distance between the ships is changing.
Therefore, dd/dt = 2x * (dx/dt) + 2y * (dy/dt) = 2 * -20 knots * 0 + 2 * 21 knots * 0 = 0.
Thus, the distance between the ships is not changing at 4 PM.
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The rate of recipt of income from the sales of vases from 1988 to 1993 can be approximated by R(t)= 100/(t+0.87)^2 billion dollars per year, where t is time in years since January 1988. Estimate to the nearest $1 billion, the total change in income from January 1988 to January 1993.
Answer choices are: $43, $53, $137, $98, $117
Answer:
The correct option is 4.
Step-by-step explanation:
It is given that the rate of recipt of income from the sales of vases from 1988 to 1993 can be approximated by
[tex]R(t)=\frac{100}{(t+0.87)^2}[/tex]
billion dollars per year, where t is time in years since January 1988.
We need to estimate the total change in income from January 1988 to January 1993.
[tex]I=\int_{0}^{5}R(t)dt[/tex]
[tex]I=\int_{0}^{5}\frac{100}{(t+0.87)^2}dt[/tex]
[tex]I=100\int_{0}^{5}\frac{1}{(t+0.87)^2}dt[/tex]
On integration we get
[tex]I=-100[\frac{1}{(t+0.87)}]_{0}^{5}[/tex]
[tex]I=-100(\frac{1}{5+0.87}-\frac{1}{0+0.87})[/tex]
[tex]I=-100(-0.979)[/tex]
[tex]I=97.9[/tex]
[tex]I\approx 98[/tex]
The total change in income from January 1988 to January 1993 is $98. Therefore the correct option is 4.
1.) Given P(E or F) = 0.82, P(E) = 0.18, and P(E and F) = 0.09, what is P(F)?
Answer:
p(F)=0.73
Step-by-step explanation:
we have by identity
[tex]p(A\cup B)=p(A)+p(B)-p(A\cap B)[/tex]
Thus for given events E and F
we have[tex]p(E\cup F)=p(E)+p(F)-p(E\cap F)[/tex]
Applying values we get
[tex]p(F)=p(E\cup F)+p(E\cap F)-p(E)[/tex]
Thus
p(F) = 0.82+0.09-0.18
p(F) = 0.73
A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $3,000 or $6,000. If the partnership raised $258,000, then how many investors contributed $3,000 and how many contributed $6,000?
Answer:
There are 26 investors which contributed 6,000
And 34 investors which contributed 3,000
Step-by-step explanation:
You need to set up a two equation problem
[tex]\left \{ {{258,000=6,000y+3,000x} \atop {60=y+x}} \right.[/tex]
Now you have to clear "x" or "y" from the second equation:
[tex]y = 60 - x[/tex]
And replace on the first equation:
[tex]258,000 = 6,000 (60 - x) + 3,000x\\258,000 = 360,000 - 6,000x + 3,000x\\3,000x = 360,000 - 258,000\\x = 102,000/3,000\\x = 34[/tex]
And now you use this "x" vale on the second equation
[tex]60 = y + x\\60 = y + 34\\60 - 34 = y\\y = 26[/tex]
There are 26 investors which contributed 6,000
And 34 investors which contributed 3,000
A tenth of a number in algebraic expression
Answer:
Step-by-step explanation: manej
Final answer:
In algebra, a tenth of a number is algebraically represented by multiplying the number by [tex]10^{-1}[/tex], which is equivalent to dividing the number by 10. This application of negative exponents simplifies expressions, especially in scientific notation, making it easier to work with large and small quantities.
Explanation:
In algebra, when we refer to a tenth of a number, we are usually dealing with fractions or exponential notation. A tenth of a number can be represented algebraically as the number divided by 10, which is the same as multiplying the number by [tex]10^{-1}[/tex]. This is because negative exponents indicate the reciprocal of a number; in other words, 10-1 equals 1/10 or 0.1.
This concept relates to the powers of ten and how each power of 10 affects the size of a number. For instance, 102 is 100, and 101 is 10, which is ten times smaller than 100. Conversely, 100 is 1, which is ten times smaller than 10, and thus, logically, [tex]10^{-1}[/tex] is 0.1, which is ten times smaller still. In expressing measurements in scientific work, especially for very small numbers, we frequently use this exponential form.
Thus, a tenth of an algebraic expression would mean multiplying the expression by [tex]10^{-1}[/tex] or dividing the expression by 10. This process is a form of simplification and re-scaling of numbers that are commonly used in scientific notation, which includes both positive and negative exponents. By understanding these principles, one can efficiently work with both large and small quantities in scientific and mathematical contexts.
Find a compact form for generating function of the sequence 1, 8, 27,.........., k^3,.........
The generating function is [tex]f(x)[/tex] where
[tex]f(x)=\displaystyle\sum_{k=0}^\infty a_kx^k[/tex]
with [tex]a_k=k^3[/tex] for [tex]k\ge0[/tex].
Recall that for [tex]|x|<1[/tex], we have
[tex]g(x)=\dfrac1{1-x}=\displaystyle\sum_{k=0}^\infty x^k[/tex]
Taking the derivative gives
[tex]g'(x)=\dfrac1{(1-x)^2}=\displaystyle\sum_{k=1}^\infty kx^{k-1}=\sum_{k=0}^\infty(k+1)x^k[/tex]
[tex]\implies g'(x)-g(x)=\dfrac x{(1-x)^2}=\displaystyle\sum_{k=0}^\infty kx^k[/tex]
Taking the derivative again, we get
[tex]g''(x)=\dfrac2{(1-x)^3}=\displaystyle\sum_{k=2}^\infty k(k-1)x^{k-2}=\sum_{k=0}^\infty(k^2+3k+2)x^k[/tex]
[tex]\implies g''(x)-3g'(x)+g(x)=\dfrac{x^2+x}{(1-x)^3}=\displaystyle\sum_{k=0}^\infty k^2x^k[/tex]
Take the derivative one last time to get
[tex]g'''(x)=\dfrac6{(1-x)^4}=\displaystyle\sum_{k=3}^\infty k(k-1)(k-2)x^{k-3}=\sum_{k=0}^\infty(k^3+6k^2+11k+6)x^k[/tex]
[tex]\implies g'''(x)-6g''(x)+7g'(x)-g(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}=\displaystyle\sum_{k=0}^\infty k^3x^k[/tex]
So the generating function is
[tex]\boxed{f(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}}[/tex]
One common system for computing a grade point average (GPA) assigns 4 points to an A, 3 points to a B, 2 points to a C, 1 point to a D, and 0 points to an F. What is the GPA of a student who gets an A in a 3-credit course, a B in each of three 4-credit courses, a C in a 3-credit course, and a D in a 2-credit course?
Answer:
2.8
Step-by-step explanation:
The weighted average is found by dividing the total number of points by the total number of credits.
GPA = (4×3 + 3×4 + 3×4 + 3×4 + 2×3 + 1×2) / (3 + 4 + 4 + 4 + 3 + 2)
GPA = 56 / 20
GPA = 2.8
The GPA of the student will be 2.8.
What is Algebra?Algebra is the study of mathematical symbols, and the rule is the manipulation of those symbols.
One common system for computing a grade point average (GPA) assigns 4 points to an A, 3 points to a B, 2 points to a C, 1 point to a D, and 0 points to an F.
Then the GPA of a student who gets an A in a 3-credit course, a B in each of three 4-credit courses, a C in a 3-credit course, and a D in a 2-credit course will be
GPA = (4×3 + 3×4 + 3×4 + 3×4 + 2×3 + 1×2) / (3 + 4 + 4 + 4 + 3 + 2)
GPA = 56 / 20
GPA = 2.8
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Solve the given linear Diophantine equation. Show all necessary work. A) 4x + 5y=17 B)6x+9y=12 C) 4x+10y=9
Answer:
A) (-17+5k,17-4k)
B) (-4+3k,4-2k)
C) No integer pairs.
Step-by-step explanation:
To do this, I'm going to use Euclidean's Algorithm.
4x+5y=17
5=4(1)+1
4=1(4)
So going backwards through those equations:
5-4(1)=1
-4(1)+5(1)=1
Multiply both sides by 17:
4(-17)+5(17)=17
So one integer pair satisfying 4x+5y=17 is (-17,17).
What is the slope for this equation?
Let's put it in slope-intercept form:
4x+5y=17
Subtract 4x on both sides:
5y=-4x+17
Divide both sides by 5:
y=(-4/5)x+(17/5)
The slope is down 4 and right 5.
So let's show more solutions other than (-17,17) by using the slope.
All integer pairs satisfying this equation is (-17+5k,17-4k).
Let's check:
4(-17+5k)+5(17-4k)
-68+20k+85-20k
-68+85
17
That was exactly what we wanted since we were looking for integer pairs that satisfy 4x+5y=17.
Onward to the next problem.
6x+9y=12
9=6(1)+3
6=3(2)
Now backwards through the equations:
9-6(1)=3
9(1)-6(1)=3
Multiply both sides by 4:
9(4)-6(4)=12
-6(4)+9(4)=12
6(-4)+9(4)=12
So one integer pair satisfying 6x+9y=12 is (-4,4).
Let's find the slope of 6x+9y=12.
6x+9y=12
Subtract 6x on both sides:
9y=-6x+12
Divide both sides by 9:
y=(-6/9)x+(12/9)
Reduce:
y=(-2/3)x+(4/3)
The slope is down 2 right 3.
So all the integer pairs are (-4+3k,4-2k).
Let's check:
6(-4+3k)+9(4-2k)
-24+18k+36-18k
-24+36
12
That checks out since we wanted integer pairs that made 6x+9y=12.
Onward to the last problem.
4x+10y=9
10=4(2)+2
4=2(2)
So the gcd(4,10)=2 which means this one doesn't have any solutions because there is no integer k such that 2k=9.
You take out a simple interest loan for $ 922 to pay for tuition. If the annual interest rate is 6 % and the loan must be repaid in 6 months, find the amount that you, the borrower, will have to repay. Round your answer to the nearest cent.
Answer:
The total amount to be repaid is equal to $949.66
Step-by-step explanation:
Simple interest is a type of interest which is usually applied on short term loans, where when a payment is made towards this kind of interest the payment first goes towards monthly interest and then the remainder is reverted towards the principal.
FORMULA FOR CALCULATING SIMPLE INTEREST =
[tex]\frac{PRINCIPAL \times RATE OF INTEREST \times TIME PERIOD}{100}[/tex]
Here principal = $922
interest rate = 6%
time period = 6 months (when made per annum it will be 6/12)
[tex]\frac{\$ 922 \times 6 \times 1}{100\times 2}[/tex]
SIMPLE INTEREST IS EQUAL TO $27.66
The total amount that is to be repaid is equal to
PRINCIPAL + SIMPLE INTEREST
= $922 + $27.66
= $949.66
show that {(1,1,0),(1,0,1),(0,1,1)} is linearly independent subset of r^3
Answer: Yes, the given set of vectors is a linearly independent subset of R³.
Step-by-step explanation: We are given to show that the following set of three vectors is a linearly independent subset of R³ :
B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} .
Since the given set contains three vectors which is equal to the dimension of R³, so it is a subset of R³.
To check the linear independence, we will find the determinant formed by theses three vectors as rows.
If the value of the determinant is non zero, then the set of vectors is linearly independent. Otherwise, it is dependent.
The value of the determinant can be found as follows :
[tex]D\\\\\\=\begin{vmatrix}1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1\end{vmatrix}\\\\\\=1(0\times1-1\times1)+1(1\times0-1\times1)+0(1\times1-0\times0)\\\\=1\times(-1)+1\times(1)+0\\\\=-1-1\\\\=-2\neq0.[/tex]
Since the determinant is not equal to 0, so the given set of vectors is a linearly independent subset of R³.
Thus, the given set is a linearly independent subset of R³.
let A={2, 4, 6, 8} and B={2, 3, 5, 7, 9} compute n(A)
Answer with explanation:
Given two sets
A={2, 4, 6, 8} and B={2, 3, 5, 7, 9}
⇒n(S)=Cardinality of a set
Means the number of distinct elements in a Set is called it's cardinal number.
→In Set A,total number of distinct elements is 4.
n(A)=4
Solve y''+2y' - 3y = 0, y(0) = 3, y'(0) = 11 Preview y(t) = |2e^(-3t)+5e^t Points possible: 1 This is attempt 3 of 3 Score on last attempt: 0. Score in gradebook: 0 License Submit
[tex]y''+2y'-3y=0 [/tex]
Second order linear homogeneous differential equation with constant coefficients, ODE has a form of,
[tex]ay''+by'+cy=0[/tex]
From here we assume that for any equation of that form has a solution of the form, [tex]e^{yt}[/tex]
Now the equation looks like this,
[tex]((e^{yt}))''+2((e^{yt}))'-3e^{yt}=0[/tex]
Now simplify to,
[tex]e^{yt}(y^2+2y-3)=0[/tex]
You can solve the simplified equation using quadratic equation since,
[tex]e^{yt}(y^2+2y-3)=0\Longleftrightarrow y^2+2y-3=0[/tex]
Using the QE we result with,
[tex]\underline{y_1=1}, \underline{y_2=-3}[/tex]
So,
For two real roots [tex]y_1\neq y_2[/tex] the general solution takes the form of,
[tex]y=c_1e^{y_1t}+c_2e^{y_2t}[/tex]
Or simply,
[tex]\boxed{y=c_1e^t+c_2e^{-3t}}[/tex]
Hope this helps.
r3t40
Express the given expanded numeral as a Hindu-Arabic numeral. (8x102) +(4x10)(2x1)
Answer:
The Hindu-Arabic numeral form of the given expanded numeral is 842.
Step-by-step explanation:
The given expanded numeral is
[tex](8\times 10^2)+(4\times 10)+(2\times 1)[/tex]
We need to express the given expanded numeral as a Hindu-Arabic numeral.
According to Hindu-Arabic numeral the given expanded numeral is written as
[tex](8\times 10^2)+(4\times 10)+(2\times 1)=(8\times 100)+(4\times 10)+(2\times 1)[/tex]
On simplification we get,
[tex](8\times 10^2)+(4\times 10)+(2\times 1)=(800)+(40)+(2)[/tex]
[tex](8\times 10^2)+(4\times 10)+(2\times 1)=842[/tex]
Therefore the Hindu-Arabic numeral form of the given expanded numeral is 842.
You invested a total of $9,000 at 4 1/2 % and 5% simple interest. During one year, the two accounts earned $435. How much did you invest in each account
Answer:
The amount invested at 4.5% was [tex]\$3,000[/tex]
The amount invested at 5% was [tex]\$6,000[/tex]
Step-by-step explanation:
we know that
The simple interest formula is equal to
[tex]I=P(rt)[/tex]
where
I is the Final Interest Value
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
Let
x -----> the amount invested at 4.5%
9,000-x -----> the amount invested at 5%
in this problem we have
[tex]t=1\ year\\ P1=\$x\\ P2=\$(9,000-x)\\I=\$435\\r1=0.045\\r2=0.05[/tex]
substitute
[tex]435=x(0.045*1)+(9,000-x)(0.05*1)[/tex]
[tex]435=0.045x+450-0.05x[/tex]
[tex]0.05x-0.045x=450-435[/tex]
[tex]0.005x=15[/tex]
[tex]x=\$3,000[/tex]
so
[tex]9,000-x=\$6,000[/tex]
therefore
The amount invested at 4.5% was [tex]\$3,000[/tex]
The amount invested at 5% was [tex]\$6,000[/tex]
Graph the function y = √ x + 4 – 2. Then state the domain and range of the function.
Answer:
Domain : [-4, ∞)
Range : [ -2, ∞)
Step-by-step explanation:
The given function is y = [tex]\sqrt{(x+4)}-2[/tex]
Domain of the given function will be
(x + 4) ≥ 0
[ Since square root of numbers less than 0 is not possible ]
Domain : x ≥ (-4)
Or [-4, ∞) will be the domain
Now range of the function will be x ≥ -2
[ -2, ∞) will be the range of the given function.
Industry standards suggest that 13 percent of new vehicles require warranty service within the first year. Jones Nissan in Sumter, South Carolina, sold 12 Nissans yesterday. (Round your mean answer to 2 decimal places and the other answers to 4 decimal places.) What is the probability that none of these vehicles requires warranty service
Answer: 0.1880
Step-by-step explanation:
The binomial distribution formula is given by :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]
where P(x) is the probability of x successes out of n trials, p is the probability of success on a particular trial.
Given : The probability of new vehicles require warranty service within the first year : p =0.13.
Number of trials : n= 12
Now, the required probability will be :
[tex]P(x=0)=^{12}C_0(0.13)^{0}(1-0.13)^{12-0}\\\\=(1)(0.13)^{0}(0.87)^{12}=0.188031682201\approx0.1880[/tex]
Thus, the probability that none of these vehicles requires warranty service = 0.1880
Any equation or inequality with variables in it is a predicate in the domain of real numbers. For the following statement, tell whether the statement is true or false. (∀x)(x4> x)
The statement is given by:
∀ x , [tex]x^4>x[/tex]
This statement is false
Since, if we consider,
[tex]x=\dfrac{1}{2}[/tex]
then we have:
[tex]x^4=(\dfrac{1}{2})^4\\\\i.e.\\\\x^4=\dfrac{1}{2^4}\\\\i.e.\\\\x^4=\dfrac{1}{16}[/tex]
Also, we know that:
[tex]\dfrac{1}{16}<\dfrac{1}{2}[/tex]
( Since, two number with same numerator; the number with greater denominator is smaller than the number with the smaller denominator )
Hence, we get:
[tex]x^4<x[/tex]
when [tex]x=\dfrac{1}{2}[/tex]
Hence, the result :
[tex]x^4>x[/tex] is not true for all x belonging to real numbers.
Hence, the given statement is a FALSE statement.
Final answer:
The statement (∀x)(x⁴ > x) is false, as it does not hold true for all real numbers. For instance, when x is a negative number like -1, the inequality x⁴ > x is false.
Explanation:
False
Explanation:
Given statement: (∀x)(x⁴ > x)
This statement asserts that for all real numbers x, x⁴ will be greater than x. However, this statement is false because it doesn't hold for all real numbers. For instance, when x is a negative number such as -1, (-1)4 is greater than -1, which means the inequality x⁴ > x is false.
An auto license plate consists of 6 digits; the first three are any letter (from the 26 alphabets), and the last three are any number from 0 to 9. For example, AAA 000, ABC 123, and ZZZ 999 are three possible license plate numbers. How many different license plate numbers may be created?
Answer: There are 17576000 ways to generate different license plates.
Step-by-step explanation:
Since we have given that
Numbers are given = 0 to 9 = 10 numbers
Number of letters = 26
We need to generate the license plate numbers.
Since there are repetition allowed.
We would use "Fundamental theorem of counting".
So, the number of different license numbers may be created as given as
[tex]26\times 26\times 26\times 10\times 10\times 10\\\\=26^3\times 10^3\\\\=17576\times 1000\\\\=17576000[/tex]
Hence, there are 17576000 ways to generate different license plates.
How does the Binomial Theorem’s use Pascal’s triangle to expand binomials raised to positive integer powers?
Answer:
There are ways for quickly multiply out a binomial that's being raised by an exponent. Like
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) = a3 + 3a2b + 3ab2 + b3
and so on and so on
but there was this mathematician named Blaise Pascal and he found a numerical pattern, called Pascal's Triangle, for quickly expanding a binomial like the ones from earlier. It looks like this
1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
Pascal's Triangle gives us the coefficients for an expanded binomial of the form (a + b)n, where n is the row of the triangle.
Hope this helps!
The Binomial Theorem uses Pascal's triangle for expanding binomials raised to positive integer powers. Pascal's triangle provides the coefficients for each term in the binomial expansion, simplifying the expansion process.
Explanation:
The Binomial Theorem uses Pascal's triangle to expand binomials that are raised to positive integer power. Pascal's triangle is a triangular array of binomial coefficients. Each line of the triangle represents the coefficients of the terms of a binomial expansion. Let's take for instance binomial expansion of (a + b)n = an+nan-1b+…from Pascal's triangle, the coefficients are 1,n, and so on. The role of Pascal's triangle in this case is pivotal in knowing the coefficients of each term in the binomial expansion and thus facilitates the expansion process.
Learn more about Binomial Theorem and Pascal's Triangle here:https://brainly.com/question/32022572
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