An interviewer is given a list of potential people she can interview. She needs five interviews to complete her assignment. Suppose that each person agrees independently to be interviewed with probability 2/3. What is the probability she can complete her assignment if the list has______.
(a) 5 names?
(b) What if it has 8 names?
(c) If the list has 8 names what is the probability that the reviewer will contact exactly 7 people in completing her assignment?
(d) With 8 names, what is the probability that she will complete the assignment without contacting every name on the list?

To find the probability that a randomly selected utility bill is between $110 and $130, we can use the formula for z-score. By calculating the z-scores for both values, we can find the areas under the curve and subtract them to get the probability.

To find the probability that a randomly selected utility bill is between $110 and $130, we can use the formula for the z-score:

z = (x - μ) / σ

where x is the value we are looking for, μ is the mean, and σ is the standard deviation.

In this case, x = $110, μ = $121, and σ = $23.

Substituting these values into the formula, we get:

z = (110 - 121) / 23 = -0.4783

Using a z-table or a calculator, we can find that the area to the left of -0.4783 is approximately 0.3186.

Next, we repeat the process for $130:

z = (130 - 121) / 23 = 0.3913

Using a z-table or a calculator, we can find that the area to the left of 0.3913 is approximately 0.6480.

To find the probability that the utility bill is between $110 and $130, we subtract the area to the left of $110 from the area to the left of $130:

P(110 ≤ X ≤ 130) = 0.6480 - 0.3186 = 0.3294

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Answer:

  0.91

Step-by-step explanation:

The total of all numbers in the diagram is 35 +5 +10 +5 = 55.

The total of the numbers inside one or both circles is 35 +5 +10 = 50.

The probability of choosing a random student from inside one or both circles (plays some instrument) is 50 out of 55, or ...

  50/55 ≈ 0.9090909... ≈ 0.91

P(A∪B) ≈ 0.91

Answer:

  y = 2^(x–2) + 3

Step-by-step explanation:

The equation above is the one that is graphed. You can pick it from the offered choices by recognizing that the horizontal asymptote on the graph is y=3. That is 3 units above the horizontal asymptote of the parent exponential function. Hence, you must have ...

  y = (some exponential) +3

_____

Please note that the exponent indicator (^) and the grouping parentheses on the exponent are essential. Without those, the equation is that of the line y=2x+1, which is not what is graphed.

Height of the pole (DC) is 57.2709m

Step-by-step explanation:

Here, Wire DA and Wire DB supports a pole.

Given that Angle, A=54 , B= 72.

Also, AB = 23m

Now, Taking triangle BCD and Using basic trigonometry

Height of pole H = DC

[tex]TanB = \frac{DC}{BC}[/tex]

[tex]BC= \frac{DC}{TanB}[/tex]

Now, Taking triangle ACD and Using basic trigonometry

[tex]TanA = \frac{DC}{AC}[/tex]

[tex]AC= \frac{DC}{TanA}[/tex]

From figure, we know that

AC = AB + BC

AC - BC = AB = 23

Replacing values of AC and BC

[tex]\frac{DC}{TanA} - \frac{DC}{TanB}=23\\

DC(\frac{1}{TanA}-\frac{1}{TanB})=23[/tex]

Now, TanB= Tan72 =3.0776 and TanA = Tan54=1.3763

[tex]DC (\frac{1}{1.3763} - \frac{1}{3.0776})_= 23[/tex]

[tex]DC ( 0.7265-0.3249)= 23[/tex]

[tex]DC ( 0.4016 )= 23[/tex]

[tex]DC = 57.2709 [/tex]

Thus, Height of thepole is 57.2709m

You have 31 nickels and 44 dimes.

Step-by-step explanation:

Total coins = 75

Worth of coins = $5.95 = 5.95*100 = 595 cents

1 nickel = 5 cents

1 dime = 10 cents

Let,

Number of nickels = x

Number of dimes = y

According to given statement;

x+y=75   Eqn 1

5x+10y=595    Eqn 2

Multiplying Eqn 1 by 5

[tex]5(x+y=75)\\5x+5y=375\ \ \ Eqn\ 3\\[/tex]

Subtracting Eqn 3 from Eqn 2

[tex](5x+10y)-(5x+5y)=595-375\\5x+10y-5x-5y=220\\5y=220[/tex]

Dividing both sides by 5

[tex]\frac{5y}{5}=\frac{220}{5}\\y=44[/tex]

Putting y=44 in Eqn 1

[tex]x+44=75\\x=75-44\\x=31[/tex]

You have 31 nickels and 44 dimes.

Keywords: linear equations, subtraction

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See the answers in explanation

Let's solve this problem as follows:

[tex]\bullet \ \frac{2x-8}{x-4} \\ \\ Common \ factor \ 2 \ from \ the \ numerator: \\ \\ \frac{2x-8}{x-4}=\frac{2(x-4)}{x-4} =2[/tex]

[tex]\bullet \ \frac{4x-8}{4x+20} \\ \\ Common \ factor \ 4 \ from \ the \ numerator \ and \ denominator: \\ \\ \frac{4x-8}{4x+20}=\frac{4(x-2)}{4(x+5)}=\frac{(x-2)}{(x+5)}[/tex]

[tex]\bullet \ \frac{x+7}{x^2+4x-21} \\ \\ Rearranging \ denominator: \\ \\ \frac{x+7}{x^2-3x+7x-21}=\frac{x+7}{x(x-3)+7(x-3)}=\frac{x+7}{x(x-3)+7(x-3)} \\ \\ Common \ factor \ x-3 \ from \ denominator: \\ \\ \frac{x+7}{(x-3)(x+7)}=\frac{1}{x-3}[/tex]

[tex]\bullet \ \frac{x^2-3x-10}{x+2} \\ \\ Rearranging \ numerator: \\ \\ \frac{x^2-3x-10}{x+2}=\frac{x^2-5x+2-10}{x+2}=\frac{x(x-5)+2(x-5)}{x+2} \\ \\ Common \ factor \ x-5 \\ \\ \frac{(x-5)(x+2)}{x+2}=x-5[/tex]

[tex]\bullet \ \frac{x^2-4}{2-x} \\ \\ Difference \ of \ squares \ from \ numerator: \\ \\ \frac{(x-2)(x+2)}{2-x} \\ \\ Common \ factor \ -1 \ from \ denominator: \\ \\ \frac{(x-2)(x+2)}{-(x-2)}=-(x+2)[/tex]

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Answer: a. 120, b. 386, c. 176, d. 968.

Step-by-step explanation:

For a combination of any number, is given as

C n,r = n!/r!(n-r)!

Please note that "n,r" is a subscript, and the exclamation mark "!" is called factorial.

From the question, n = 10

EXACTLY 3 0s

10 combination 3

r is exactly 3, that is equal 3.

C 10,3= 10!/3!(10-3)! = 10!/3!7!= 120.

For clarification,

10!/3!7!=10×9×8/3×2×1 = 120.

You can also use a calculator to compute the factorials.

MORE 0s than 1s

There will be more 0s than 1s when < 5bits are 0s.

We have r<5

Therefore for r=4

C 10,4 = 10!/4!(10-4)!=10!/4!6!=210

r=3

C 10,3= 10!/3!(10-3)!=10!/3!7!=120

r=2

C 10,2=10!/2!(10-2)!=10!/2!8!=45

r=1

C 10,1=10!/1!(10-1)!=10!/1!9!=10

r=0

C 10,0=10!/0!(10-0)!=10!/0!10!=1

Summing the answers gives us our final answer

210+120+45+10+1= 386.

AT LEAST 7 1s

To get this combination, the value of r will be greater than or equal to 7

r>=7

We have,

r=7

C 10,7=10!/7!(10-7)!=10!/7!3!=120

r=8

C 10,8=10!/8!(10-8)!=10!/8!2!=45

r=9

C 10,9=10!/9!(10-9)!=10!/9!1!=10

r=10

C 10,10=10!/10(10-10)!=10!/10!0!=1

120+45+10+1= 176

AT LEAST 3 1s

the value for r will be greater than or equal to 3:

We can the values of r from 3 to 10.

r=3

C 10,3=10!/3!(10-3)!=120

r=4

C 10,4=10!/4!(10-4)!=10!/4!6!=210

r=5

C 10,5=10!/5!(10-5)!=10!/5!5!=252

r=6

C 10,6=10!/6!(10-6)!=10!/6!4!=210

r=7

C 10,7=10!/7!(10-7)!=10!/7!3!=120

r=8

C 10,8=10!/8!(10-8)!=10!/8!2!=45

r=9

C 10,9=10!/9!(10-9)!=10!/9!1!=10

r=10

C 10,10=10!/10!(10-10)!=10!/10!0!=1

Adding our answers gives 968.

The bits can be either 1 or 0. The total number of bit string for each specified case is:

Since the items are indistinguishable, their arrangements doesn't matter.

They can be chosen in [tex]^nC_r = \dfrac{n.(n-1).(n-2)...(n-(r+2)).(n-(r+1))}{r.(r-1).(r-2)...3.2.1} \: \rm (r \leq n)[/tex]

The bit string is of length 10.

Each bit can be in one of the two states, viz 0 or 1.

Evaluating the count of bit strings for given cases:

Think of it as if there are 10 seats and 3 people to sit on. They're going to be 0s. 3 seats can be chosen from 10 seats in [tex]^{10}C_3 = \dfrac{10\times 9\times 8}{3 \times 2\times 1} = 120[/tex] ways.

The three 0s are identical, so no intra-arrangement between them matters.

Thus, total 120 such strings exist.

It means, 0s can be 6,7,8,9, or 10 places.

Just similar to above case, 0s on x places out of 10 places can be in [tex]^{10}C_x[/tex] ways.

Thus, total such strings of 0s being more than 1s and being 10 bit strings are:

[tex]^{10}C_6 + ^{10}C_7 + ^{10}C_8 + ^{10}C_9 + ^{10}C_{10} =210+120+45+10+1=386[/tex]

At least seven 1s means either 7, 8, 9, or 10 ones.

Total count of such strings are:

[tex]^{10}C_7 + ^{10}C_8 + ^{10}C_9 + ^{10}C_{10} =120+45+10+1=176[/tex]

They are three or more ones. Total count of such strings is:

[tex]^{10}C_3 + ^{10}C_4 + ^{10}C_5+^{10}C_6 + ^{10}C_7 + ^{10}C_8 + ^{10}C_9 + ^{10}C_{10} =120 + 210 + 252 + 210+120+45+10+1=386+582=968[/tex]

Thus, the total number of bit string for each specified case is:

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Answer:

a) 100000

b) 90000

Step-by-step explanation:

We have the possibility of 10 digits

(0,1,2,3,4,5,6,7,8,9)

If there are no restrictions on the digit, there are 10 possibilities for each of the five digits

We then have;

10*10*10*10*10

= 10^5

= 100000

This means 100000 zip codes are possible if there are no restrictions.

b) If the first digit cannot be 3, there are 9 possibilities. This is because the possibility of the first digit being 3 is 1 out of 10. Therefore the possibility of not being 3 is 9 out of 10

The other four digits have 10 possibilities each.

So we have

9*10*10*10*10

= 90000

This means there are 90000 zip codes if the first digit does not start with 3

Answer: c

Step-by-step explanation:

The minimum sample size does not depend on the size of the population

The size of the population, N, is NOT required to determine the minimum sample size for estimating a population mean, contrasting with the required elements like the desired confidence level, margin of error, and population standard deviation.

The question asks which factor is NOT required to determine the minimum sample size needed to estimate a population mean. The options are:

The correct answer is C. The size of the​ population, N. When estimating a population mean, the key factors required include the desired confidence level, the desired margin of error, and the value of the population standard deviation (sigma), but not necessarily the size of the population. This is especially true in cases where the population is very large or infinite, and the sample size needed for a specific confidence level and margin of error can be calculated without this information.

Answer:

the answer is 3

7x + 9= 30

7x = 21

x = 3

Answer:

x = 3

Step-by-step explanation:

Collect like terms;

7x + 9 = 30

Subtract 9 from both sides;

7x = 21

Divide both sides by 7;

x = 3

Answer:

Step-by-step explanation:

The quadrilateral has 4 sides and only two of them are equal.

A) to find PR, we will consider the triangle, PRQ.

Using cosine rule

a^2 = b^2 + c^2 - 2abcos A

We are looking for PR

PR^2 = 8^2 + 7^2 - 2 ×8 × 7Cos70

PR^2 = 64 + 49 - 112 × 0.3420

PR^2 = 113 - 38.304 = 74.696

PR = √74.696 = 8.64

B) to find the perimeter of PQRS, we will consider the triangle, RSP. It is an isosceles triangle. Therefore, two sides and two base angles are equal. To determine the length of SP,

We will use the sine rule because only one side,PR is known

For sine rule,

a/sinA = b/sinB

SP/ sin 35 = 8.64/sin110

Cross multiplying

SPsin110 = 8.64sin35

SP = 8.64sin35/sin110

SP = (8.64 × 0.5736)/0.9397

SP = 5.27

SR = SP = 5.27

The perimeter of the quadrilateral PQRS is the sum of the sides. The perimeter = 8 + 7 + 5.27 + 5.27 = 25.54 cm

Answer:

  p(x) = x(x +5)(x -9)

Step-by-step explanation:

If r is a root, then (x -r) is a factor of the polynomial. For the given roots, the factorization is ...

  p(x) = (x -0)(x -(-5))(x -9)

  p(x) = x(x +5)(x -9)

To find the difference between the number of tetras and minnows, set up a system of equations using the given ratios. Solve the system to find the number of tetras, guppies, and minnows. Finally, subtract the number of minnows from the number of tetras to find the difference.

To determine the difference between the number of tetras and minnows in the fish tank, we need to first find the number of each type of fish. We can do this by setting up a system of equations using the given ratios. Let T represent the number of tetras, G represent the number of guppies, and M represent the number of minnows.

From the first ratio, we have T/G = 4/2. Simplifying this equation, we get T = 2G.

From the second ratio, we have M/G = 1/3. Simplifying this equation, we get M = (1/3)G.

Since we know there are a total of 60 fish in the tank, we can create the equation T + G + M = 60. Substituting the previous equations into this equation, we get 2G + G + (1/3)G = 60. Solving for G, we find G = 9. Plugging this value into the equations for T and M, we get T = 2(9) = 18 and M = (1/3)(9) = 3.

Therefore, there are 18 tetras and 3 minnows in the fish tank. The difference between the number of tetras and minnows is 18 - 3 = 15.

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Answer: (39.424, 61.576)

Step-by-step explanation:

When population standard deviation([tex]\sigma[/tex]) unknown ,The confidence interval for population mean is given by :-

[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]

, where n= Sample size

[tex]\overline{x}[/tex] = sample mean.

s= sample standard deviation

[tex]t^*[/tex] = Critical t-value (two-tailed)

Given : n= 15

Degree of freedom= 14  [df=n-1]

[tex]\overline{x}=\ $50.50[/tex]

[tex]s^2=400\\\\\Rightarrow\ s=\sqrt{400}=20[/tex]

Significance level = [tex]\alpha=1-0.95=0.05[/tex]

For [tex]\alpha=0.05[/tex] and df = 14, the critical t-values : [tex]t^*=\pm2.1448[/tex]

Then the 95% confidence interval for population mean will be  :

[tex]50.50\pm (2.1448)\dfrac{20}{\sqrt{15}}\\\\=50.50\pm(2.1448)(5.1640)\\\\\approx50.50\pm11.076\\\\=(50.50-11.076,\ 50.50+11.076)\\\\=(39.424,\ 61.576)[/tex]

Hence, a 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain's new store in the mall. : (39.424, 61.576)

The 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain's new store in the mall is calculated using the sample mean ($50.50), sample size (15), sample standard deviation (20), and Z-value for a 95% confidence interval (1.96). The calculated interval is (-$1.11, $102.11).

To construct a 95% confidence interval for the average amount that the department store's credit card customers spent on their first visit to their new store, we would use the formula for a confidence interval:

CI = X-bar ± (Z-value * (s/√n)),

where X-bar is the sample mean = $50.50, n is the sample size = 15, s is the sample standard deviation = √400 = 20, and Z-value is the critical value from the Z-table which, for a 95% confidence interval, equals 1.96.

Plug these values into the formula,

CI = 50.5 ± (1.96 * (20/√15))

Using a calculator, the confidence interval comes out to (-$1.11, $102.11).

So, we are 95% confident that the average amount its credit card customers spent on their first visit to the chain's new store in the mall lies between $-1.11 and $102.11.

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Priya practiced the violin for 114 minutes

Given that  

Han spent 75 minutes practicing the piano over the weekend

Priya practiced the violin for 152% as much as Han practiced the piano

Need to determine how long did priya practice  

Duration of Han practicing the piano over weekend = 75 minutes

As given that Priya practiced the violin for 152% as much as Han practiced the piano

=> Duration of Priya practicing the piano = 152% of Han practicing the piano

=> Duration of Priya practicing the piano = 152% of 75 minutes

We know that a % of b is written in fraction as [tex]\frac{a}{100} \times b[/tex]

[tex]\Rightarrow \text { Duration of Priya practicing the piano }=\frac{152}{100} \times 75=114 \text { minutes }[/tex]

Hence Priya practiced the violin for 114 minutes

Jacob's survey is a study in statistics, specifically looking at the correlation between the amount of time spent on sports and student's mood ratings. A line is fit to the data to determine the relationship, with the direction of the line offering insights into how these two variables correlate, but this does not imply causation.

From your question, Jacob performed a survey asking about the number of hours students spent playing sports in the past day and asked them to rate their mood. It's a study of basic statistics, specifically focusing on correlation between two variables, here those are the number of hours spent on sports and mood ratings. To determine a relationship between these variables, a line is often fit to the data, using methods like linear regression.

For example, if the line on the graph is rising, it indicates a positive correlation between the amount of sports played and a student's mood, meaning that as sports playtime goes up, so does mood ratings. A falling line means there's a negative correlation. If there's no clear direction, it's likely that there's no significant correlation between the two variables. But remember that correlation doesn't mean causation: just because two things correlate doesn't mean that one causes the other.

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Answer:

  c. uses the average of the most recent data values in the time series as the forecast for the next period.

Step-by-step explanation:

We assume you want to complete the description of moving averages method.

The moving in moving averages refers to the fact that the data points used to compute the average are some number of most recent data points. As data is accumulated, the data used to compute the average "moves" to include the newest data and exclude the oldest data.

Answer:

Step-by-step explanation:

Set this up according to the Triangle Proportionality Theorem:

[tex]\frac{3x}{4x}=\frac{3x+7}{5x-8}[/tex]

Cross multiply to get

[tex]3x(5x-8)=4x(3x+7)[/tex]

and simplify to get

[tex]15x^2-24x=12x^2+28x[/tex]

Get everything on one side of the equals sign and solve for x:

[tex]3x^2-52x=0[/tex] and

[tex]x(3x-52)=0[/tex]

By the Zero Product Property,

x = 0 or 3x - 52 = 0 so x = 17 1/3

Answer:

z= -0.968

We can conclude that we fail to reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that  they would watch one of the television shows not differs from 0.5 or 50% .  

Step-by-step explanation:

1) Data given and notation n  

n=106 represent the random sample taken

X=48 represent the people who says that  they would watch one of the television shows.

[tex]\hat p=\frac{48}{106}=0.453[/tex] estimated proportion of people who says that  they would watch one of the television shows.

[tex]p_o=0.5[/tex] is the value that we want to test

[tex]\alpha[/tex] represent the significance level  

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that 50% of people who says that  they would watch one of the television shows.:  

Null hypothesis:[tex]p=0.5[/tex]  

Alternative hypothesis:[tex]p \neq 0.5[/tex]  

When we conduct a proportion test we need to use the z statisitc, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

3) Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.453 -0.5}{\sqrt{\frac{0.5(1-0.5)}{106}}}=-0.968[/tex]  

4) Statistical decision  

P value method or p value approach . "This method consists on determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level is not provided, but we can assume [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z<-0.968)=0.333[/tex]  

So based on the p value obtained and using the significance level assumed [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we fail to reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that  they would watch one of the television shows not differs from 0.5 or 50% .  

deltax = (56-0)/4 = 14

number of intervals 4

The intervals are:

(0,14),(14,28),(28,42),(42,56)

The Midpoint Rule =

f(7)+f(21)+f(35)+f(49)

[0.47577]+[-0.99159]+[-0.35892]+[0.65699]

deltax =14

sum = -0.21774

Multiplying by deltax = -3.0484

To approximate the integral of sin(x^2) dx from 0 to 5 using the Midpoint Rule with n=5, first calculate Δx = 1. Next, perform calculations with x values 0.5, 1.5, 2.5, 3.5, and 4.5. Finally, use the Midpoint Rule formula to calculate the approximated integral.

To solve this problem, you'll need to use the Midpoint Rule, which is a numerical method used to approximate definite integrals. In this case, we want to approximate ∫ from 0 to 5 of sin(x^2) dx with an n value of 5.

The Midpoint Rule can be represented as: Mn = Δx[f(x1) + f(x2) + ... + f(xn)], where Δx = (b-a)/n and each xi = a + (Δx/2) + (i-1)Δx.

Here, we'll first find our Δx = (5-0)/5 = 1, then calculate each xi and plug those into our function. Our xi values will be 0.5, 1.5, 2.5, 3.5, and 4.5. Finally, we plug into our formula and solve to find our approximated integral value.

A detailed and step-by-step solution would involve calculating each f(xi) with the xi values given above, and then adding these up

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Answer:

Step-by-step explanation:

Answer:

Average rowing velocity of boat in still water is 4 miles per hour and average velocity of the current is 1 mile per hour.

Step-by-step explanation:

We are given the following in the question:

Let x be the average rowing velocity of boat in still water and y be the the average velocity of the current.

[tex]\text{Speed} = \displaystyle\frac{\text{Distance}}{\texr{Time}}[/tex]

The boat rowed 7.5 miles​ downstream, with the​ current, in 1.5 hours.

Velocity with the current =

[tex]=\text{average rowing velocity of boat in still water} + \text{ average velocity of the current} = x + y[/tex]

Thus, we can write the equation:

[tex]7.5 = (x+y)1.5\\x+y = 5[/tex]

The return trip​ upstream, against the​ current, covered the same​ distance, but took 2.5 hours.

Velocity against the current =

[tex]=\text{average rowing velocity of boat in still water} - \text{ average velocity of the current} = x - y[/tex]

Thus, we can write the equation:

[tex]7.5 = (x-y)2.5\\x-y = 3[/tex]

Solving, the two equations:

[tex]2x = 8\\x = 4, y = 1[/tex]

Thus, average rowing velocity of boat in still water is 4 miles per hour and average velocity of the current is 1 mile per hour.

Answer:

At x = 1 and maximum area  = 0.0499

Step-by-step explanation:

The hypotenuse of a right triangle has one end at the origin and other end on the curve, [tex]y=x^2e^{-3x}[/tex]  with x ≥ 0.

One leg of right triangle is x-axis and another leg parallel to y-axis.

Length of base of right triangle =  x

Height of right triangle = y

Area of right triangle, [tex]A=\dfrac{1}{2}xy[/tex]

[tex]A=\dfrac{1}{2}x^3e^{-3x}[/tex]

For maximum/minimum value of area.

[tex]\dfrac{dA}{dx}=\dfrac{3}{2}x^2e^{-3x}-\dfrac{3}{2}x^3e^{-3x}[/tex]

Now, find critical point, [tex]\dfrac{dA}{dx}=0[/tex]

[tex]\dfrac{3}{2}x^2e^{-3x}-\dfrac{3}{2}x^3e^{-3x}=0[/tex]

[tex]\dfrac{3}{2}x^2e^{-3x}(1-x)=0[/tex]

x =0,1

For x = 0, y = 0

For x = 1, [tex]y=e^{-3}[/tex]

using double derivative test:-

[tex]\dfrac{d^2A}{dx^2}=\dfrac{6}{2}xe^{-3x}-\dfrac{9}{2}x^2e^{-3x}-\dfrac{9}{2}x^2e^{-3x}-\dfrac{9}{2}x^3e^{-3x}[/tex]

At x= 0 , [tex]\dfrac{d^2A}{dx^2}=0[/tex]

Neither maximum nor minimum

At x = 1, [tex]\dfrac{d^2A}{dx^2}=-0.14<0[/tex]

Maximum area at x = 1

The maximum area of right triangle at x = 1

Maximum area, [tex]A=\dfrac{1}{e^3}\approx 0.0499[/tex]

The point of maxima will be x=3 and the maximum area will be 0.002 square units.

According to the diagram attached

The area of the given triangle will be = 0.5*base*height

As one end of the hypotenuse is on the curve [tex]y = x^2e^(-3x)[/tex], Coordinates of one end of the hypotenuse will be [tex](x, x^2e^(-3x)[/tex].

Area A(x) of the given triangle = 0.5*base*  height

Base =  x

Height = [tex]x^2e^(-3x)[/tex]

So A(x) = [tex]0.5*x*x^{2} *e^(-3x)[/tex]

[tex]A(x) = 0.5*x*x^{2} *e^(-3x)\\\\A(x) = 0.5 x^3e^(-3x)[/tex]

For the maximum area,

[tex]A'(x) = 0\\\\x^2e^(-3x) (x-3) = 0\\x = 0 and x=3[/tex]will be the points of extremum.

Points of extremum are the values of x for which a function f(x) attains a maximum or minimum value.

A(0) = 0

A(3) = 0.002

Therefore, The point of maxima will be x=3, and the maximum area will be 0.002 square units.

To get more about maxima and minima visit:

https://brainly.com/question/6422517

Answer:

15.87% of the invoices were paid within 15 days of receipt

Step-by-step explanation:

An invoice was paid an average of 20 days after it was received.

Mean = [tex]\mu = 20[/tex]

Standard deviation = [tex]\sigma = 5[/tex]

Now we are supposed to find what percent of the invoices were paid within 15 days of receipt i.e.P(x<15)

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

At x = 15

Substitute the values

[tex]Z=\frac{15-20}{5}[/tex]

[tex]Z=-1[/tex]

Refer the z table for p value

So, p value = 0.1587

So, 15.87% of the invoices were paid within 15 days of receipt

Answer:

The required probability is [tex]\frac{3}{4}[/tex].

Step-by-step explanation:

Consider the provided information.

There are 8 leaders.

Thus, the total number of ways to arrange 8 leaders are 8!.

Assume that Obama and Berlusconi is one person.

Therefore the total number of leaders are 7 (As Obama and Berlusconi is one person).

The number of ways in which 7 leader can be arranged: 7!

Although Obama and Berlusconi is one unit but they can interchange their place in 2 ways. Like Obama and Berlusconi or Berlusconi and Obama

That means the total number of ways : 7!×2

The number of ways in which they are not next to each other = Total number of ways - The number of ways in which they are next to each other

Number of ways they are not next to each other = 8!-7!×2

The probability that they are not next to each other = [tex]\frac{8!-7!\times2}{8!}=\frac{3}{4}[/tex]

Hence, the required probability is [tex]\frac{3}{4}[/tex].

Answer:

A factor

Step-by-step explanation:

Take the equation 2 x 4 = 8 as an example.

2 and 4 are multiplied together to get 8.

2 and 4 are factors, and 8 is the product.

Answer:

B) 2 hours

Step-by-step explanation:

If machine  A complete a job in 3 1/2 hours or 7/2 of an hour

means that in one hour finished 1÷ 7/2     or  2/7

If machine  B complete a job in 4 2/3 hours or  14/3 of an hour

means that in one hour finished 1÷ 14/3    or  3/14 of an hour

Then the two machines working together in one hour will make

2/7 + 3/14    =  (4 + 3)/ 14

or   7/14   = 1/2

half of the job. Therefore these two machines working together will take two hours

Answer: The speed of Bob is 56.8 km/hr.

Step-by-step explanation:

Let the speed of Bob be 'x'.

Let the speed of John be 'x-8'.

Distance covered = 230 miles

time = [tex]1\dfrac{1}{2}=\dfrac{3}{2}\ hr[/tex]

According to question, we get that

[tex]\dfrac{230}{x}-\dfrac{230}{x+8}=\dfrac{3}{2}\\\\230\dfrac{x+8-x}{x(x+8)}=\dfrac{3}{2}\\\\\dfrac{230\times 8}{x^2+8x}=\dfrac{3}{2}\\\\\dfrac{1840}{x^2+8x}=\dfrac{3}{2}\\\\1840\times 2=3x^2+24x\\\\3680=3x^2+24x\\\\3x^2+24x-3680=0\\\\x\approx 56.8\ km/hr[/tex]

Hence, the speed of Bob is 56.8 km/hr.

Answer:

f(g(x)) = g(f(x))  = x and f and g are the inverses of each other.

Step-by-step explanation:

Here, the given functions are:

[tex]f(x) = x^2 - 3, g(x) = \sqrt{({3+x)} }[/tex]

To Show:  f (g(x))  = g (f (x))

(1)  f (g(x))

Here, by the composite function:

[tex]f (g(x)) = f (\sqrt{3+x} )  = \sqrt{(3+x)} ^2 - 3  =  (3 + x) - 3  =  x[/tex]

⇒ f (g(x))  = x

(2) g (f(x))

Here, by the composite function:

[tex]g(f(x)) = g(x^2 -3)   = \sqrt{3 +(x^2 -3) }  = \sqrt{x^2}   = x[/tex]

⇒ g (f(x))  = x

Hence, f(g(x)) = g(f(x))  = x

⇒ f and g are the inverses of each other.