Answer:
Infinite circles but with values (h,k,r)=(h,3h-3,√(10h²-30h+25))
Step-by-step explanation:
You are correct, if we only have two points, then we will have infinite possibilities, but all those circles will have to be attached to those two points. We have to find a condition to have infinite circles which pass through those points.
Let h be the x coordinate of the center of the circle
Let k be the y coordinate of the center of the circle
Let r be the radius of the circle. Having the circle equation:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
And now expand every binomial and substitute the values of the points:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}\\\\(0-h)^{2}+(2-k)^{2}=r^{2}\\\\(3-h)^{2}+(1-k)^{2}=r^{2}[/tex]
[tex]x^{2}-2xh+h^{2}+y^{2}-2yk+k^{2}=r^{2}\\\\h^{2}+4-4k+k^{2}=r^{2} (I)\\\\9-6h+h^{2}+1-2k+k^{2}=r^{2}[/tex]
Now substract the first and secod equation, and then the first and the third one:
[tex]x^{2}-2xh+y^{2}-4-2yk+4k=0\\\\x^{2}-9-2xh+6h+y^{2}-1-2yk+2k=0[/tex]
Now we substract the second equation from the first one:
[tex]-4+4k+9-6h+1-2k=0[/tex]
Reduce terms and we will have a linear equation, clear k:
[tex]2k-6h+6=0\\\\k=3h-3[/tex]
The last equation let us know the value of k if we asign any value to h.
Now we substitute h and k=3h-3 in I:
[tex]h^{2}+4-4(3h-3)+(3h-3)^{2}=r^{2}\\\\10h^{2}-30h+25=r^{2}\\\\r=\sqrt{10h^{2}-30h+25}[/tex]
The last equation let us know the value of r if we asign any value to h.
These equations (in fact functions), will let us know every set for h, k and r to form a circle (which will pass through the mentioned points)
[tex](h,k,r)=(h,3h-3,\sqrt{10h^{2}-30h+25})[/tex]
Where the indepent variable is h for any real, k is any real and the minimum r is √2.5:
[tex]h:R\\k:R\\r:(\sqrt{2.5},infinite)[/tex]
two automobiles leave a city at the same time and travel along straight highways that differ in direction by 84 degrees. if their speeds are 60 mi/hr and 45 mi/hr, approximatly how far apart are the cars at the end of 20 minutes?
Answer:
D = 23.7 miles
Step-by-step explanation:
Given data:
[tex]\theta=84 degrees[/tex]
Time[tex] = 20 min = \frac{1}{3} hr[/tex]
Distance of A [tex]= (60 mi/hr)\times \frac{1}{3} hr =20 mi[/tex]
Distance of B =[tex] (45 mi/hr)\frac{1}{3} hr = 15 mi[/tex]
Draw a triangle.
By using cosine formula we can determine the distance between them
[tex]D^2 = 20^2 + 15^2 - 2\times 20\times 15\times cos(84)[/tex]
[tex]D^2 = 625 - 600 cos(84) [/tex]
[tex]D^2 = 625 - 63.2 [/tex]
[tex]D^2 = 561.8[/tex]
Thus D = 23.7 miles
A bird flies from its nest 9 km in the direction of 60 degrees north of east, where it stops to rest on a tree. It then flies 20 km in the direction due southeast and lands atop a telephone pole. Place an xy-coordinate system so that the origin is the bird's nest, the x-axis points east, and the y-axis points north. a. At what point is the tree located? b. At what point is the telephone pole located?
Answer with explanation:
The motion of the bird is shown in the attached figure
part a)
In the ordinate axis as shown in the attached figure we can see that the tree is located at a point [tex](9cos(60),9sin(60)=(4.5km,7.794km)[/tex] with respect to origin O
Part b)
The location of the telephone pole as seen from the co-ordiante axis is at
[tex]x=4.5+20cos(45)=18.64km\\\\y=7.794-20sin(45)=-6.348km[/tex]
The points are located as shown in the figure
Customers arrive at a bank drive-up window every 6 minutes based on a Poisson distribution. Once they arrive at the teller, service time is exponentially distributed based on a rate of 5 minutes per transaction,a.What is the probability that 3 or fewer customers will arrive in one hour?b.How many customers are most likely in line (waiting) at any one point in time?c.What will be the average time waiting in the system?d.What will be the average time in the system?
Answer:
(a) 0.01029 (b) 4.167 customers (c) 0.4167 hours or 25 minutes (d) 0.5 hours or 30 minutes
Step-by-step explanation:
With an arrival time of 6 minutes, λ=10 clients/hour.
With a service time of 5 minutes per transaction, μ=12 transactions/hour.
(a) The probability of 3 or fewer customers arriving in one hour is
P(C<=3)=P(1)+P(2)+P(3)
[tex]P(X)=\lambda^{X}*e^{-\lambda} /X![/tex]
[tex]P(1)=10^{1}*e^{-10} /1!=0.00045\\P(3)=10^{3}*e^{-10} /3!=0,00227\\P(2)=10^{2}*e^{-10} /2!=0,00757\\P(x\leq3)=0.00045+0.00227+0.00757 = 0.01029[/tex]
(b) The average number of customers waiting at any point in time (Lq) can be calculated as
[tex]L_{q}=p*L=\frac{\lambda}{\mu}*\frac{\lambda}{\mu-\lambda}\\L_{q}=\frac{10}{12}*\frac{10}{12-10}\\\\L_{q}=100/24=4.167[/tex]
(c) The average time waiting in the system (Wq) can be calculated as
[tex]W_{q}=p*W=\frac{\lambda}{\mu}*\frac{1}{\mu-\lambda}\\W_{q}=(10/12)*(1/(12-10))\\W_{q}=(10/24)=0.4167[/tex]
(d) The average time in the system (W), waiting and service, can be calculated as
[tex]W=\frac{1}{\mu-\lambda}\\W=\frac{1}{12-10}=0.5[/tex]
Freedonia has 49 senators. Each senator is either honest or corrupt. Suppose you knowthat at least two of the Freedonian senators are honest and that, out of any threeFreedonian senators, at least one is corrupt. Based on these facts, can you determinehow many Freedonian senators are honest and how many are corrupt? if so, what is theanswer?
Answer:
There are 47 corrupt senators and 2 honest senators.
Step-by-step explanation:
Freedonia has 49 senators. Each senator is either honest or corrupt.
At least 2 of the Freedonian senators are honest and that, out of any 3 Freedonian senators, at least 1 is corrupt.
As we can see that there are no 3 senators where all of them are honest. So either there is one senator who is corrupt and 2 are honest.
Also it is given that at least two of the Freedonian senators are honest.
Hence, we can conclude there are 2 honest senators and 47 corrupt senators.
the scale of a map is 1 cm : 72km. What is the actual distance between two towns that are 4 cm apart on the map
If 2^n + 1 is an odd prime for some integer n, prove that n is a power of 2. (H
Step-by-step explanation:
We will prove by contradiction. Assume that [tex]2^n + 1[/tex] is an odd prime but n is not a power of 2. Then, there exists an odd prime number p such that [tex]p\mid n[/tex]. Then, for some integer [tex]k\geq 1[/tex],
[tex]n=p\times k.[/tex]
Therefore
[tex]2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p.[/tex]Here we will use the formula for the sum of odd powers, which states that, for [tex]a,b\in \mathbb{R}[/tex] and an odd positive number [tex]n[/tex],
[tex]a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-...+b^{n-1})[/tex]
Applying this formula in 1) we obtain that
[tex]2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p=(2^k+1)(2^{k(p-1)}-2^{k(p-2)}+...-2^{k}+1)[/tex].
Then, as [tex]2^k+1>1[/tex] we have that [tex]2^n+1[/tex] is not a prime number, which is a contradiction.
In conclusion, if [tex]2^n+1[/tex] is an odd prime, then n must be a power of 2.
Which is traveling faster, a car whose velocity vector is 201 + 25), or a car whose velocity vector is 30i, assuming that the units are the same for both directions? is the faster car. At what speed is the faster car traveling? speed = Enter the speed as a NUMBER, using exact values or at least 4 decimal place accuracy.
Answer with explanation:
For any object having the velocity vector as
[tex]\overrightarrow{v}=v_x\widehat{i}+v_y\widehat{j}+v_z\widehat{k}[/tex]
the magnitude of velocity is given by
[tex]|v|=\sqrt{v_x^2+v_y^2+v_z^2}[/tex]
For car 1 the velocity vector is
[tex]\overrightarrow{v}_1=20\widehat{i}+25\widehat{j}[/tex]
Therefore
[tex]|v_1|=\sqrt{20^2+25^2}\\\\\therefore v_1=32.0156units[/tex]
Similarly for car 2 we have
[tex]\overrightarrow{v}_2=30\widehat{i}[/tex]
Therefore
[tex]|v_2|=\sqrt{30^2}\\\\\therefore v_2=30.0units[/tex]
Comparing both the values we find that car 1 has the greater speed.
The correct answer is that the car with velocity vector 30i is traveling faster, and its speed is 30 units.
To determine which car is traveling faster, we need to calculate the magnitude (or speed) of each velocity vector. The magnitude of a velocity vector in two dimensions is given by the square root of the sum of the squares of its components.
For the first car, the velocity vector is[tex]\( \mathbf{v}_1 = 20\mathbf{i} + 25\mathbf{j} \). The magnitude of this vector is calculated as follows:\[ ||\mathbf{v}_1|| = \sqrt{(20)^2 + (25)^2} = \sqrt{400 + 625} = \sqrt{1025} \][/tex]
For the second car, the velocity vector is [tex]\( \mathbf{v}_2 = 30\mathbf{i} + 0\mathbf{j} \).[/tex]
The magnitude of this vector is calculated as:[tex]\[ ||\mathbf{v}_2|| = \sqrt{(30)^2 + (0)^2} = \sqrt{900 + 0} = \sqrt{900} = 30 \]Now, comparing the magnitudes of the two vectors:\[ ||\mathbf{v}_1|| = \sqrt{1025} \approx 32.0156 \]\[ ||\mathbf{v}_2|| = 30 \]It is clear that \( ||\mathbf{v}_1|| \) is approximately 32.0156 units, while \( ||\mathbf{v}_2|| \)[/tex] is exactly 30 units. Since 32.0156 is greater than 30, the car with velocity vector[tex]\( \mathbf{v}_2 = 30\mathbf{i} \)[/tex] is traveling faster.Therefore, the faster car is traveling at a speed of 30 units.
Can any one answer this please
Answer: 10.6
Step-by-step explanation:
divide the number she spent (2.65) by the amount the store charges per ounce (.25)
2.65/ .25 = 10.6
If f(x)=8x then what is the area enclosed by the graph of the function, the horizontal axis, and vertical lines at x=2 and x=6
Answer:
[tex]A=128[/tex]
Step-by-step explanation:
First of all we need to graph f(x)=8x, (First picture)
Now we have to calculate the area enclosed by the graph of the function, the horizontal axis, and vertical lines at [tex]x_{1}=2[/tex] and [tex]x_{2}=6[/tex] ,
The area that we have to calculate is in pink (second picture).
The function is positive and the domain is [tex][2,6][/tex] then we can calculate the area with this formula:
[tex]A=\int\limits^b_a {f(x)} \, dx[/tex],
In this case [tex]b=x_{2} , a=x_{1}[/tex]
[tex]A=\int\limits^6_2 {8x} \, dx = 8\int\limits^6_2 {x} \, dx[/tex]
The result of the integral is,
[tex]A=8\frac{x^{2}}{2}[/tex], but the integral is defined in [2,6] so we have to apply Barrow's rule,
Barrow's rule:
If f is continuous in [a,b] and F is a primitive of f in [a,b], then:
[tex]\int\limits^b_a {f(x)} \, dx =F(b)-F(a)[/tex]
Applying Barrow's rule the result is:
[tex]A=8.\frac{6^{2} }{2}-8.\frac{2^{2} }{2}[/tex]
[tex]A=8.\frac{36}{2} -8.\frac{4}{2}[/tex]
[tex]A=144-16[/tex]
[tex]A=128[/tex]
Which of the following items would not be used for geometric probability? a. area of a rug c. length of time b. number of trees d. length of a field
Answer: b. number of trees
Step-by-step explanation:
The concept of geometric probability is basically use when we have continuous data .
Since it is impossible to count continuous data , but geometrically ( in form of length, area etc) we can count the outcomes in general to calculate the required probability.
Therefore, from the given options , Option b. "number of trees" would not be used for geometric probability because among all it is the only discrete case which is countable.
Rest of items ( a. area of a rug , c. length of time , d. length of a field) would be used for geometric probability,
A bag contains three red marbles, two green ones, one lavender one, four yellows, and five orange marbles. HINT [See Example 7. How many sets of four marbles include all the red ones?
Answer:
There are 12 sets of four marbles include all the red ones.
Step-by-step explanation:
Given : A bag contains three red marbles, two green ones, one lavender one, four yellows, and five orange marbles.
To find : How many sets of four marbles include all the red ones?
Solution :
Number of red marbles = 3
Number of green marbles = 2
Number of lavender marbles = 1
Number of yellow marbles = 4
Number of orange marbles = 5
We have to form sets of four marbles include all the red ones,
For position of getting red ones we have three red marbles i.e. [tex]^3C_3[/tex]
For the fourth one we have 12 choices i.e. [tex]^{12}C_1[/tex]
Total sets of four marbles include all the red ones is
[tex]=^3C_3\times ^{12}C_1[/tex]
[tex]=1\times 12[/tex]
[tex]=12[/tex]
Therefore, There are 12 sets of four marbles include all the red ones.
Greg is covering a wall with equal-sized tiles that cannot be cut
into smaller pieces. The wall is 66 inches high by 72 inches wide.
What is the largest square tile that simon can use?
A. 9 in.
B. 16 in.
C. 36 in.
D. 64 in.
Answer:
C. 36 in.
Step-by-step explanation:
The wall is 66 inches high by 72 inches wide.
The tiles are
a) 9 inch square = 3 by 3
b) 16 inch square = 4 by 4
c) 36 inch square = 6 by 6
d) 64 inch square = 8 by 8
Factors of 66 = 2 x 3 x 11
Factors of 72 = 2 x 2 x 2 x 3 x 3
Now, we can see that in both 66 and 72 , we have 2 x 3 common that is 6.
And square of 6 is 36.
So, the answer is option C.
A car has mass 1500 kg and is traveling at a speed of 35 miles/hour. what is its kinetic energy in joules? (Be sure to convert miles/hour to m/s). If the car increases its speed to 70 miles/hour, by what factor does its kinetic energy increase? show work
Answer:
The kinetic energy of car with mass 1500 kg and with speed of 35 miles/hour is KE=183598 J and when the car increases its speed to 70 miles/hour the kinetic energy changes by a factor of 4.
Step-by-step explanation:
The first step is to convert the speed miles/hour to m/s.
[tex]35\frac{miles}{hour} *\frac{1609.34 \>m}{1 \>miles}*\frac{1 \>hour}{3600 \> s}=15.646 \frac{m}{s}[/tex]
Next, the formula for the kinetic energy is
[tex]KE=\frac{1}{2} mv^{2}[/tex]
So input the values given:
[tex]KE=\frac{1}{2} (1500)(15.646)^{2}\\KE=750 \cdot (15.646)^{2}\\KE=183597.987 = 183598 \frac{kg \cdot \>m^{2}}{s^{2}} \\KE=183598 \>J[/tex]
Notice that the speed of 70 miles/hour is the double of 35 miles/hour so we can say that [tex]v_{2}=2v_{1}[/tex] and use the formula for the kinetic energy
[tex]KE_{2} =\frac{1}{2} m(v_{2}) ^{2}\\if \: v_{2}= 2v_{1}, then \:\\KE_{2} =\frac{1}{2} m(2v_{1}) ^{2}\\KE_{2} =\frac{1}{2} m4(v_{1})^{2}\\KE_{2} =4(\frac{1}{2} m(v_{1})^{2})\\We \:know \:that \:KE_{1} =\frac{1}{2} m(v_{1})^{2} so\\KE_{2} =4(KE_{1})[/tex]
We can see that when the car increases its speed to 70 miles/hour the kinetic energy changes by a factor of 4.
A linear revenue function is R = 12x. (Assume R is measured in dollars.) what is the slope? What is the revenue received from selling one more item if 50 are currently being sold?
Answer:
12 ; 12 dollars
Step-by-step explanation:
Data provided in the question:
Revenue function, R = 12x
R is in dollars
Now,
The slope can be found out by differentiating the above revenue function w.r.t 'x'
thus,
[tex]\frac{\textup{dR}}{\textup{dx}}[/tex]= [tex]\frac{\textup{d(12x)}}{\textup{dx}}[/tex]
or
slope = 12
Now, for the second case of selling one more unit i.e x = 1, the revenue can be obtained by substituting x = 1 in revenue function
therefore,
R = 12 × 1 = 12 dollars
An opinion poll contacts 1101 adults and asks them, " Which political party do you think has better ideas for leading the country in the 21st century?" In all, 613 of the 1101 say, " The Democrats." The sample in this setting is
A. the 613 people who chose the Democrats.
B. the 1101 people interviewed.
C. all 235 million adults in the United States.
D. None of the above.
Answer:
B. the 1101 people interviewed.
Step-by-step explanation:
An opinion poll contacts 1101 adults and asks them, " Which political party do you think has better ideas for leading the country in the 21st century?"
The sample in this setting is the 1101 people interviewed.
Here sample space is 1101.
Answer:
B. the 1101 people interviewed.
Step-by-step explanation:
The sample is the subset of the population and it takes for making the experiment easy. Further, the sample is said to be best if it is the representation of the whole population.
Here, since opinion poll contacts all 1101 adults for knowing their opinion about political parties.
Hence the sample space is all 1101 people interviewed.
A pharmacist weighed 475 mg of a substance on a balance of dubious accuracy. When checked on a balance of high accuracy, the weight was found to be 445 mg. Calculate the percentage error in the first weighing.
Answer: 6.416%
Step-by-step explanation:
The percentage error formula is given by :-
[tex]\%\text{error}=\dfrac{|\text{Estimate-Actual}|}{\text{Actual}}\times100[/tex]
Given : The estimated weight of a substance = 475 mg
The actual weight of the substance = 445 mg
Then,
[tex]\%\text{error}=\dfrac{| 475-445|}{445}\times100\\\\=\dfrac{30}{445}\times100=6.74157303371\approx6.416\%[/tex]
Hence, the percentage error in the first weighing. = 6.416%
Final answer:
The percentage error in the pharmacist's first weighing is approximately 6.74%, calculated by subtracting the accurate weight from the inaccurate weight, resulting in an absolute error of 30 mg, and then dividing the absolute error by the accurate weight, multiplying by 100.
Explanation:
To calculate the percentage error of the pharmacist's initial weighing, we first need to determine the absolute error by subtracting the accurate weight from the inaccurate weight. In this instance, the initial weight ( A ) recorded was 475 mg, and upon checking with a high accuracy balance, the true weight ( B ) was found to be 445 mg. Therefore, the absolute error ( Δ ) is the difference between these two measurements: Δ = A - B = 475 mg - 445 mg = 30 mg.
After determining the absolute error, we can calculate the percentage error using the following formula:
Percentage Error = ( Δ / B ) × 100%
Substituting in the respective values, we get:
Percentage Error = (30 mg / 445 mg) × 100% ≈ 6.74%
Give an example of either Inductive or Deductive reasoning. Try to use a "real world" example of something you might have encountered or found on the internet. Your example could involve numbers, or some other kind of scientific information. If you have trouble coming up with an example, just use "Google" for some ideas.
Answer:
See answer below
Step-by-step explanation:
In inductive reasoning we arrive to a general conclusion based on particular observations.
For example:
"My friends Peter and Mary study at XYZ University. Peter and Mary are brilliant.
Yesterday I met Joe in a party. Joe also studies at XYZ University and he happens to be brilliant, too.
I conclude that all the students of XYZ University are brilliant"
Notice that this kind of reasoning might lead to a false conclusion.
In deductive reasoning, we arrive to a particular conclusion based on general observations. So, deductive reasoning is the opposite of inductive reasoning.
An example of deductive reasoning could be the following:
"To be accepted in the XYZ University you must pass a test with a score greater than 70%.
My friend Peter studies at XYZ University, so he passed the test with a score greater than 70%"
In deductive reasoning you may arrive to a false conclusion if your general assumption is false.
Answer:
but did now am the am
Step-by-step explanation:
An aircraft seam requires 30 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. (Round your answers to four decimal places.) (a) If 21% of all seams need reworking, what is the probability that a rivet is defective?
Answer:
[tex] 0.0078[/tex]
Step-by-step explanation:
To compute the probability of a rivet being defective we can do the following:
The seam won't need reworking if the 30 rivets are working as intended. Since there's a 21% chance of the seam needing reworking, we then know that there's a 79% chance of the seam not needing reworking, which means that there's a 79% chance of having the 30 rivets working as intended. Now, each rivet is either defective with a probability p, or NOT defective with a probability 1-p. Since rivets are defective independently from one another, the probability of the 30 rivets working as intended is [tex] (1-p)^{30}[/tex], and since we know this has a chance of happening of 79%, we get the equation:
[tex] 0.79=(1-p)^{30}[/tex]
Solving for p, we get:
[tex] 0.79^{1/30}=1-p[/tex]
[tex] p=1-0.79^{1/30}\approx 0.0078[/tex]
300 ml over 15 min is what rate of ml per hour
Answer:
1200 ml per hour
Step-by-step explanation:
To compute what rate is 300 ml over 15 mins as a rate of ml per hour, we do a rule of 3, using a variable x as that amount of ml we don't know yet. We should have everything in the same units, so instead of writing 1 hour we write 60 minutes:
[tex] \frac{300~ml}{15~min}=\frac{x~ml}{60~mins}[/tex]
Now we solve for x:
[tex] \frac{300~ml}{15~min}\cdot 60~mins=x~ml[/tex]
[tex] \frac{18000~ml}{15} =x~ml[/tex]
[tex] 1200~ml =x~ml[/tex]
And so, now that we know the value of x, the rate we wanted to find is
[tex]\frac{1200~ml}{60~mins}[/tex]
Which is just 1200 ml per hour.
On a True/False test, Amy answered the first three questions wrong but answered the rest of the questions correctly. On the same test, Scott answered exactly two questions wrong. He answered the last question wrong, the 26 questions before it correctly, and the question before that wrong. If Amy and Scott were both incorrect on the same question exactly once, what is the greatest possible percent of the total number of questions that Amy could have answered correctly?
Final answer:
Amy could have answered approximately 89.29% correctly, assuming the test had the minimum number of questions allowing for all conditions in the problem to be satisfied.
Explanation:
The question involves determining the greatest possible percent of questions Amy could have answered correctly on a true/false test, given specific details about how Amy and Scott answered their questions.
To find the maximum percentage Amy could have gotten correct, we must assume that the test had the smallest number of questions that fulfill all the given conditions. This would be when the question before the last question, which Scott got wrong, aligns with one of the first three questions that Amy got wrong. Since they were both incorrect only once on the same question, Amy's third question is the same as Scott's second-to-last question. Thus, the test would have a minimum of 28 questions (the 26 Scott got right, plus the two he got wrong).
Amy answered the first three questions wrong and all others correctly, so she got 25 out of 28 questions correct. To find the percentage, we divide 25 by 28 and multiply by 100. Therefore, Amy answered approximately 89.29% of the questions correctly. Amy could not have a higher percentage because having more questions on the test would only lower her score.
Begin by clearly stating the problem; you can copy this from the problem set file. Show how you solved the problem step by step, including any formulas or equations that you used. Finally, state your final answer in a complete sentence, and make sure that you include the correct units with you answer.
1.)Albert borrowed $19,100 for 4 years. The simple interest is $9932.00. Find the rate.
Answer: 13%
Step-by-step explanation:
We know that the formula to find the simple interest :
[tex]I=Prt[/tex], where P is the principal amount , r is rate (in decimal )and t is the time period (in years).
Given : Albert borrowed $19,100 for 4 years. The simple interest is $9932.00. Find the rate.
i.e. P = $19,100 and t= 4 years and I = $9932.00
Now, Substitute all the values in the formula , we get
[tex]9932=(19100)r(4)\\\\\Rightarrow\ r=\dfrac{9932}{19100\times4}\\\\\Rightarrow\ r=0.13\ \ \ \text{[On simplifying]}[/tex]
In percent, [tex]r=0.13\times100\%=13\%[/tex]
Hence, the rate of interest = 13%
Consider the claim: If m is an even integer, then m^2+5m-1 is an odd integer.
(a.) Prove the claim using a direct proof.
(b.) State the converse. Is the converse true? Provide a proof or give a counterexample.
I have read the previous examples and am having trouble with them so please don't copy and paste a retired answer.
Rewriting the expression using m=2p we have:
Answer:
[tex]m^{2} +5 -1[/tex] is an odd integer but the converse is not true.
Step-by-step explanation:
Even numbers are written as 2n where n is any integer, while odd numbers are written as 2n-1 where n is any integer.
a) To prove that [tex]m^{2} +5m-1[/tex] is an odd integer, we have to prove that it can be written as 2n-1.
By hypothesis, m is an even integer so we will write it as 2p.
Rewriting the original expression using [tex]m=2p[/tex] we have:
[tex]m^{2} +5m-1 = (2p)^{2} +5(2p)-1[/tex]
Solving the expression and factorizing it we get
[tex]4p^{2} +10p -1 = 2(2p^{2}+5p) -1\\ \\[/tex]
And this last expression is an expression of the form 2n-1, and therefore [tex]m^{2} +5m-1[/tex] is an odd integer.
b) The converse would be: if [tex]m^{2} +5m-1[/tex] is an odd integer, then m is an even integer.
We'll give a counterexample, let's make [tex]m=3[/tex], then
[tex]m^{2} +5m-1[/tex]
[tex]3^{2}+5(3)-1 = 23[/tex] is an odd integer but m is odd.
Therefore, the converse is not true.
Select all of the answers below that are equal to B = {John, Paul, George, Ringo, Pete, Stuart}
Question 1 options: {flowers, computer monitor, flag, teddy bear, bread, thermostat}
{bookmark, needle, street lights, sock, greeting card, Ringo}
{The Monkees} {book, door, speakers, soap, toothpaste, pool stick}
{scotch tape, iPod, Sharpie, Street Lights, window, clock}
{Paul, Ringo, Pete, John, George, Stuart}
Answer:
The correct option is 5) {Paul, Ringo, Pete, John, George, Stuart}.
Step-by-step explanation:
Consider the provided sets:
B = {John, Paul, George, Ringo, Pete, Stuart}
Two sets are equal if all the elements of Sets are same.
Set B has the elements: John, Paul, George, Ringo, Pete and Stuart
Now consider the provided options of sets.
From the provided options of set only option 4 has all the elements of set B but the order is different.
Thus, the correct option is 5) {Paul, Ringo, Pete, John, George, Stuart}.
720 divided by 7? I'm so confused..
Answer:
102.8571 :)
Step-by-step explanation:
Answer:102.857 or rounded to 103
Step-by-step explanation:
You divide 720 by 7 which = 102.857
If it asks for a rounded number it would be 103
What is the negation of the following: "If n is divisible by 6 then n is divisible by both 2 and 3."
A. n is divisible by 6 or n is divisible by both 2 and 3.
B. If n is divisible by both 2 and 3 then n is divisible by 6.
C. If n is divisible by 6 then n is divisible by both 2 and 3.
D. n is divisible by 6 and n is not divisible by both 2 and 3.
E. If n is not divisible by 6 then n is not divisible by both 2 and 3.
F. n is not divisible by 6 or n is divisible by both 2 and 3.
G. n is divisible by 6 and n is divisible by both 2 and 3.
H. n is not divisible by 6 and n is divisible by both 2 and 3.
I. If n is not divisible by 6 then n is divisible by both 2 and 3.
J. If n is not divisible by both 2 and 3 then n is not divisible by 6.
K. n is divisible by 6 or n is not divisible by both 2 and 3.
Answer:
D. n is divisible by 6 and n is NOT divisible by both 2 and 3.
Step-by-step explanation:
The statement we want to negate is:
"IF n is divisible by 6 THEN n is divisible by both 2 and 3"
you should think of it as having one single antecedent (the sentence that follows after the IF) and one single consequent (the sentence that follows after THEN), as:
"IF ( n is divisible by 6 )THEN ( n is divisible by both 2 and 3 )"
This kind of statements are negated by saying that the antecedent is true, but the consequent isn't true (which is kind of saying that the antecedent being true doesn't necessarily make the consequent also true).
So the negation of the original statement is just:
"(n is divisible by 6) and NOT (n is divisible by both 2 and 3)"
which in common english is just
"n is divisible by 6 and n is NOT divisible by both 2 and 3."
Answer:
hippioo
Step-by-step explanation:
hippi hipppokmkmkouse
if one pie is shared equally by 6 people, then each person will get
what fraction of the pie?
Answer: Each person will take [tex]\dfrac{1}{6}[/tex] of the pie.
Step-by-step explanation:
Given : One pie is shared equally by 6 people.
The total number of persons = 6
Now, the fraction of the whole pie each person will take :-
[tex]\dfrac{\text{Number of pie}}{\text{total persons}}\\\\=\dfrac{1}{6}[/tex]
Therefore, the fraction of the whole pie each person will take= [tex]\dfrac{1}{6}[/tex]
Trig: A sector of a circle has area 25 cm2 and centralangle
0.5 radians. Find its radius and arc length.
Answer: Radius = 10 cm and Arc length = 5 cm
Step-by-step explanation:
The area of a sector with radius r and central angle [tex]\theta[/tex] (In radian) is given by :-
[tex]A=\dfrac{1}{2}r^2\theta[/tex]
Given : A sector of a circle has area [tex]25 cm^2[/tex] and central angle 0.5 radians.
Let r be the radius , then we have
[tex]25=\dfrac{1}{2}r^2(0.5)\\\\\Rightarrow\ r^2=\dfrac{2\times25}{0.5}\\\\\Rightarrow\ r^2=\dfrac{50}{0.5}=100\\\\\Rightarrow\ r=\sqrt{100}=10\ cm[/tex]
Thus, radius = 10 cm
The length of arc is given by :-
[tex]l=r\theta=10\times0.5=5\ cm[/tex]
Hence, the length of the arc = 5 cm
Final answer:
To find the radius and arc length of a sector of a circle, we use formulas related to the circumference and central angle of a circle. The radius of the sector is 10 cm and the arc length is 5 cm.
Explanation:
To find the radius and arc length of a sector of a circle, we need to use the formulas related to the circumference and central angle of a circle. The formula for the area of a sector is given by:
Area = (θ/2) * r^2
where θ is the central angle and r is the radius of the circle. We are given that the area of the sector is 25 cm^2 and the central angle is 0.5 radians. Setting up this equation, we get:
25 = (0.5/2) * r^2
Simplifying, we find:
r^2 = 100
Taking the square root of both sides, we find:
r = 10 cm
To find the arc length, we use the formula:
Arc Length = θ * r
Substituting the values, we find:
Arc Length = 0.5 * 10 = 5 cm
Jemer lost his ball and decided to buy a new one. the new ball cost $300. which is three times the price of his old ball. how much did he pay for the old ball.
Answer:
The cost of the old ball was $100.
Step-by-step explanation:
The cost of the new ball = $300
The new ball has three times the price of his old ball.
So, let the price of the old ball be = x
As per situation, we get the equation:
[tex]3x=300[/tex]
Dividing both sides by 3:
[tex]\frac{3x}{3}= \frac{300}{3}[/tex]
=> x = 100
Hence, the cost of the old ball was $100.
ƒ(1) = 65,536, ƒ(n) = √ __ ƒ(n - 1)
Use the "at least once" rule to find the probabilities of the following events. Getting at least one tail when tossing nine fair coins The probability is (Type an integer or a simplified fraction.)
Answer: [tex]\dfrac{511}{512}[/tex]
Step-by-step explanation:
We know that the total number of faces on a coin [Tail, Heads] =2
The probability of getting a tail = [tex]\dfrac{1}{2}=\dfrac{1}{2}[/tex]
The probability of getting no tail = [tex]1-\dfrac{1}{2}=\dfrac{1}{2}[/tex]
The "at least once" rule says that the when a coin is tossed n times , then the probability of getting at least one tail is given by :-
[tex]\text{P(Atleast one tail)}=1-(\text{P(No tail)})^n[/tex]
Since , n=9
Then, the probability of getting at least one tail is given by :-
[tex]\text{P(Atleast one tail)}=1-(\text{P(No tail)})^9\\\\=1-(\dfrac{1}{2})^9\\\\=1-\dfrac{1}{512}\\\\=\dfrac{511}{512}[/tex]
The probability is [tex]\dfrac{511}{512}[/tex]
The probability of getting at least one tail when tossing nine fair coins is 511/512. This is computed by finding the complement of getting no tails (only heads) when flipping the coins.
Explanation:In this problem, we are looking for the probability of getting at least once a tail when tossing nine fair coins. To solve it, we use the rule of complementary probability. Rather than finding the probability for one or more tails, it's easier to find the probability of the complement, which is getting no tails (only heads). With a fair coin, the probability of getting a head on one toss is 1/2, so for nine tosses, it's (1/2)^9 = 1/512. However, this is the probability of getting no tails at all. We want the exact opposite; at least one tail. So, we subtract this probability from 1: P(at least one tail) = 1 - P(no tails) = 1 - 1/512 = 511/512.
Learn more about Probability here:https://brainly.com/question/32117953
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