On a coordinate plane, triangle A B C has points (negative 2, 7), (negative 2, 3), and (negative 6, 3) and triangle D E F has points (negative 2, negative 10), (negative 2, negative 2), and (6, negative 2).
Given that StartFraction A B Over D E EndFraction = StartFraction B C Over E F EndFraction = one-half, complete the statements to show that △ABC ~ △DEF by the SAS similarity theorem.

Horizontal and vertical lines are
congruent
.

So, angles
are right angles by definition of perpendicular lines.

All right angles are
.

Therefore, △ABC ~ △DEF by the SAS similarity theorem.

To find the general expression for the elasticity of demand (e_p), we need to differentiate the demand function with respect to price (p) and multiply it by the ratio of price to quantity (p/q). The elasticity of demand measures the responsiveness of quantity demanded to changes in price.

The general expression for elasticity of demand (e_p) can be calculated as:

e_p = (dQ/dp) * (p/Q)

Where dQ/dp represents the derivative of the demand function with respect to price, and Q represents the quantity demanded.

The elasticity of demand helps us understand how sensitive the quantity demanded is to changes in price. If e_p is greater than 1, demand is considered elastic, meaning that quantity demanded is highly responsive to price changes. If e_p is less than 1, demand is inelastic, indicating that quantity demanded is less responsive to price changes.

In conclusion, the general expression for the elasticity of demand (e_p) is calculated by taking the derivative of the demand function with respect to price and multiplying it by the ratio of price to quantity. This measure helps determine the responsiveness of quantity demanded to changes in price.

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Answer: 2,506 people

Step-by-step explanation:

786 + 908 + 812 = 2,506 people

The test score of the student who joined the class is 55.

To find the test score of the student who joined the class, we can use the formula for calculating the mean:

Mean = (Sum of all values) / (Number of values)

We know that the mean test score of the original 12 students was 42. This means that the sum of their test scores was:

Sum of scores = Mean x Number of students = 42 x 12 = 504

Now, when the new student joins the class, the mean test score becomes 43. This means that the sum of all 13 students' test scores is:

Sum of scores = Mean x Number of students = 43 x 13 = 559

We can subtract the sum of the original 12 students' test scores from the sum of all 13 students' test scores to find the test score of the student who joined the class:

Test score of new student = Sum of all scores - Sum of original scores
Test score of new student = 559 - 504
Test score of new student = 55

Therefore, the test score of the student who joined the class is 55.

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APR is obtained by dividing the finance charge for the loan by the total amount borrowed, given by this formula APR = ((F / P) x 12) x 100

The formula for APR (annual percentage rate) is given as;

APR = ((F / P) x 12) x 100

Where;

The annual percentage rate (APR) of a loan if you know the number of monthly payments and the finance charge per $100, is calculated as follows;

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For an individual with a femur length of 49 centimeters, we can expect their height to be approximately 177.15 centimeters.

When f=49, plugging it into the formula h=62.6+2.35f, we get h=62.6+2.35(49)=177.15.

This means that for an individual with a femur length of 49 centimeters, we would expect their height to be approximately 177.15 centimeters.

This provides an estimate of the individual's height based on the relationship between femur length and height indicated by the formula. It's important to note that this is an estimate and individual variation may exist.

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Answer:

1/6

Step-by-step explanation:

By calculations, Colleen's photo is smaller than Ali's photo

From the question, we have the following parameters that can be used in our computation:

Area of Ali's photo = 91 square inches.

For Colleen's photo, we have

9 inches by 7 inches

This means that

Area of Colleen's photo = 9 * 7 square inches.

Evaluate

Area of Colleen's photo = 63 square inches.

63 square inches is lesser than 91 square inches

This means that Colleen's photo is smaller than Ali's photo

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The estimated population proportion is 0.622, with a margin of error of +/- 0.096.

The value of the population proportion of people from Hawaii who exercised for at least 30 minutes a day 3 days a week can be estimated using the sample proportion of 62.2%. However, we need to calculate the margin of error to determine a range in which the true population proportion is likely to fall.

Using the formula for the margin of error:

Margin of error = z*sqrt(p*(1-p)/n)

where z is the z-score for the desired level of confidence (let's use 95% confidence, which corresponds to a z-score of 1.96), p is the sample proportion (0.622), and n is the sample size (100).

Plugging in the values, we get:

Margin of error = 1.96*sqrt(0.622*(1-0.622)/100) = 0.096

So the margin of error is 0.096, meaning that we can be 95% confident that the true population proportion of people from Hawaii who exercise for at least 30 minutes a day 3 days a week falls within a range of 0.622 +/- 0.096.

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The function S(t) = 31,500(1.034)^t approximates the number of digital subscriptions for a large city newspaper after the launch of a new advertising campaign, with an initial number of 31,500 subscriptions and a growth rate of 3.4% per month.

To interpret the parameters of the function S(t) = 31,500(1.034)^t, which approximates the number of digital subscriptions for a large city newspaper after the launch of a new advertising campaign.

1. The initial number of digital subscriptions (S(0)): This is represented by the constant 31,500 in the equation. When t=0 (at the launch of the campaign), the function becomes S(0) = 31,500(1.034)^0 = 31,500. This means that at the start of the advertising campaign, there were 31,500 digital subscriptions.

2. The growth rate of digital subscriptions: This is represented by the factor 1.034 in the equation. The growth rate is 3.4% (since 1.034 = 1 + 0.034).

This means that the number of digital subscriptions is expected to increase by 3.4% each month after the launch of the advertising campaign.

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The option of bank B investment will allow to reach goal of buying the car the fastest compare to bank A it will take approximately 8.4 years

Saved amount = $15,000

Value of the car = $26,795

Use the formula for compound interest to calculate the time it will take to reach $26,795 for each bank,

Bank A,

Initial investment ' P ' = $15,000

Annual interest rate ' r ' =5.45%

                                       =  0.0545

compounded continuously  'n'  = infinity

Target amount  'A' = $26,795

The formula for continuous compounding is,

A = P[tex]e^{rt}[/tex]

Substituting the values, we get,

⇒ 26,795 = 15,000 × [tex]e^{(0.0545t)}[/tex]

Solving for t, we get,

⇒ t = (log(26,795/15,000))/(0.0545)

     ≈ 9.4 years

It will take approximately 9.4 years to reach the target amount if we invest in Bank A.

Bank B,

Initial investment ' P ' = $15,000

Annual interest rate ' r ' =5.75%/12

                                       =  0.00479 (monthly interest rate)

n = 12 compounded monthly

Target amount 'A' = $26,795

The formula for monthly compounding is,

A = [tex]P\times( 1+ r/n)^{nt}[/tex]

Substituting the values, we get,

26,795 = [tex]15,000 \times(1+0.00479/12)^{12t}[/tex]

Solving for t, we get,

t = (1/12) × (log(26,795/15,000))/(log(1+0.00479/12))

  ≈ 8.4 years

It will take approximately 8.4 years to reach the target amount if we invest in Bank B.

Therefore, investment in Bank B will allow you to reach your goal of buying the car the fastest, taking approximately 8.4 years.

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The volume of the smaller cone is approximately [tex]5.5 cm^3[/tex], rounded to the nearest tenth

If the cones are similar, then the ratio of the corresponding dimensions of the cones is the same.

Let's denote the height and radius of the smaller cone as h and r, respectively. Then, the height and radius of the larger cone are 3h and 2r, respectively.

Since the volumes of the cones are proportional to the cube of their radii and heights, we can write:

(volume of smaller cone) / (volume of larger cone) = [tex](r^2 * h) / ((2r)^2 * 3h)[/tex]

Simplifying this expression, we get:

(volume of smaller cone) / (volume of larger cone) = 1/12

Since we are given that the volume of the larger cone is [tex]66 cm^3[/tex], we can solve for the volume of the smaller cone as follows:

(volume of smaller cone) =[tex](1/12) * (66 cm^3) = 5.5 cm^3[/tex]

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Answer:

v = -64

Step-by-step explanation:

First, you add 4 to both sides to isolate the variable term:

v/8 = -8

Next, you multiply both sides by 8 to isolate the variable on one side:

v = -64

So, the solution to the equation v/8 - 4 = -12 is v = -64.

The coordinates of the point prior to the dilation are (-2,-1) when the Scale factor is 1/3 and the new coordinates are (-6,3).

To find the coordinates of the point prior to the dilation, we need to use the formula for dilation:

(x’, y’) = (k x, ky)

where

(x’, y’) = the new coordinates

(x, y) = original coordinates

k = scale factor

Given data:

Scale factor =  1/3

New coordinates = (-6, 3)

By substuting the values in the equation we get:

(-6, 3) = (k x, ky)

Solving for x and y:

k x = -6

ky = 3

Dividing the ky equation by the k x  equation we get:

y/x = 3/-6

y/x = -1/2

From the above equation, we can assume that x = 2 and y = -1.

Therefore, the coordinates of the point prior to the dilation are (-2,-1).

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C(12) represents the total cost (in dollars) of renting the industrial cleaning unit for 12 hours.

The problem gives us a function C(t) = 60 + 24t, where t represents the number of hours that an industrial cleaning unit is rented for. The function tells us that the total cost (in dollars) of renting the unit is equal to $60 plus $24 per hour.

Now, we are asked to find what C(12) represents. To do so, we substitute t = 12 into the function, which gives us:

C(12) = 60 + 24(12)

We can simplify this expression by multiplying 24 by 12, which gives us:

C(12) = 60 + 288

Adding 60 and 288 together, we get:

C(12) = 348

So, C(12) represents the total cost (in dollars) of renting the industrial cleaning unit for 12 hours. Therefore, the correct answer to the question is: The cost of renting the unit for 12 hours.

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the determinant of the matrix is e1e4-e3e2

The determinant of a matrix is a scalar value that is a function of the entries. It characterizes some properties of the matrix and the linear map represented by it. The determinant is nonzero if and only if the matrix is invertible and an isomorphism exists.

Determinants are only defined for square matrices and encode certain properties of the matrices.

The determinant of a matrix is defined by the difference betweern the product of the right diagonal to the the product of the left diagonal

From the given question. the determinant of the matrix is e1*e4 -e3-e2 = e1e4-e3e2

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The nearest year, we can estimate that the number of new cases would reach 1282 in the year 2017.

To find the linear regression equation, we need to use the formula:

y = a + bx

where y is the number of new cases, x is the number of years since 1995, a is the y-intercept and b is the slope of the line.

Using the given data, we can find the values of a and b using the formulas:

b = (nΣxy - ΣxΣy) / (nΣ[tex]x^2[/tex] - (Σx)[tex]^2)[/tex]

a = (Σy - bΣx) / n

where n is the number of data points, Σxy is the sum of the products of x and y, Σx is the sum of x, Σy is the sum of y, and Σ[tex]x^2[/tex] is the sum of squares of x.

Using these formulas and the given data, we get:

n = 9

Σx = 36

Σy = 7386

Σx^2 = 162

Σxy = 3330

b = (93330 - 367386) / (9*162 - 36^2) ≈ -75.44

a = (7386 - (-75.44)*36) / 9 ≈ 2612.67

Therefore, the linear regression equation is:

y ≈ 2612.67 - 75.44x

To estimate the year in which the number of new cases would reach 1282, we can substitute y = 1282 into the equation and solve for x:

1282 ≈ 2612.67 - 75.44x

75.44x ≈ 2612.67 - 1282

x ≈ 22.36

This means that the number of new cases would reach 1282 approximately 22.36 years after 1995. Adding this to 1995 gives us an estimate of the calendar year:

1995 + 22.36 ≈ 2017.36

Rounding to the nearest year, we can estimate that the number of new cases would reach 1282 in the year 2017.

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These values (x4, y4) represent the coordinates of the missing vertex of Lana's square.

To determine the missing vertex of Lana's square, we need to know the location of the three given vertices. Let's assume that Lana plotted the vertices in a clockwise direction, starting from the top left.

If we denote the coordinates of the first vertex as (x1, y1), the second vertex as (x2, y2), and the third vertex as (x3, y3), then the coordinates of the missing vertex can be found as follows:

1. Calculate the distance between the first and second vertices:

d1 = sqrt((x2 - x1)^2 + (y2 - y1)^2)

2. Calculate the distance between the second and third vertices:

d2 = sqrt((x3 - x2)^2 + (y3 - y2)^2)

If the square is regular (i.e., all sides are of equal length), then d1 = d2. Otherwise, the shape is not a square.

3. Calculate the midpoint between the first and second vertices:

xm = (x1 + x2) / 2

ym = (y1 + y2) / 2

4. Calculate the vector that goes from the midpoint to the third vertex:

vx = x3 - xm

vy = y3 - ym

5. Calculate the coordinates of the missing vertex by adding the vector to the midpoint:

x4 = xm + vx

y4 = ym + vy

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Step-by-step explanation:

x = sqrt (y-9)     square both sides

x^2 = y-9           add 9 to both sides

y = x^2 + 9        <====== this parabola has a POSITIVE x^2 coefficient ( +1)...

                                      so it is bowl shaped and opens UPWARD

Thus, the amount after the compounding is found to be $14,683.82.

Compound interest is, to put it simply, interest that is earned on interest. Compound interest is interest that is earned on both the initial principal and interest that builds up over time in a savings account.

There may be a difference in the timing of when interest is paid out and compounded. For instance, interest on a savings account may be paid monthly but compounded daily.

Given data:

principal P: $5,000,

annual interest r: 6%,

n interest periods: 12,

number of years t : 18

Formula:

A = P[tex](1 + r/n)^{nt}[/tex]

Put the values:

A = 5000[tex](1 + 0.06/12)^{12*18}[/tex]

A = 5000*2.93

A = 14,683.82

Thus, the amount after the compounding is found to be $14,683.82.

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Complete question:

principal: $5,000, annual interest: 6%, interest periods: 12, number of years: 18

After 18 years, the investment compounded what will be worth $___.?

Answer:

The y intercept of the scatterplot is 250 sweat shirts and the slope is 10/3

What is a linear function?

y = mx + b

where m is the rate of change and b is the y intercept.

Let y represent the number of sweat shirts sold and x represent the temperature.

The y intercept is at (0,250).

Using point (0,250) and (14,200):

Slope = (200-250) / (15-0) = 10/3

The y intercept of the scatter plot is 250 sweat shirts and the slope is (10/3)

Lamar can spend at most $3.01 on a single pair of socks to stay within his budget.

a. The total cost of the socks and shorts can be represented by the expression:

Total cost = Cost of shorts + Cost of 5 pairs of socks

= $19.95 + 5x

where x is the price of one pair of socks.

b. To write an equation that says Lamar spent exactly $35 on the socks and shorts, we can equate the total cost expression to $35:

$19.95 + 5x = $35

To solve for x, we can first subtract $19.95 from both sides:

5x = $15.05

Then, divide both sides by 5:

x = $3.01

So, Lamar spent $19.95 + 5($3.01) = $35 on the socks and shorts.

c. Other possible prices for the socks that would still allow Lamar to stay within his budget of $35 can be found by plugging in values of x that satisfy the inequality:

Cost of 5 pairs of socks = 5x ≤ $15.05

For example, if the socks cost $2.99 per pair, then the total cost would be:

$19.95 + 5($2.99) = $34.90

which is within Lamar's budget.

d. We can write an inequality to represent the amount Lamar can spend on a single pair of socks as:

x ≤ (35 - 19.95)/5

This simplifies to:

x ≤ $3.01

So, Lamar can spend at most $3.01 on a single pair of socks to stay within his budget.

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Answer: -10.57

Step-by-step explanation:

Answer:

0.25 years

Step-by-step explanation:

Penelope invested $89,000 in an account paying an interest rate of 6⅜% compounded continuously.

To calculate the time it would take Penelope's money to double, use the continuous compounding interest formula.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]

As the principal amount is doubled, then A = 2P.

Given interest rate:

Substitute A = 2P and r = 0.06375 into the continuous compounding interest formula and solve for t:

[tex]\implies 2P=Pe^{0.06375t}[/tex]

[tex]\implies 2=e^{0.06375t}[/tex]

[tex]\implies \ln 2=\ln e^{0.06375t}[/tex]

[tex]\implies \ln 2=0.06375t\ln e[/tex]

[tex]\implies \ln 2=0.06375t(1)[/tex]

[tex]\implies \ln 2=0.06375t[/tex]

[tex]\implies t=\dfrac{\ln 2}{0.06375}[/tex]

[tex]\implies t=10.872896949...[/tex]

Therefore, it will take 10.87 years for Penelope's investment to double.

[tex]\hrulefill[/tex]

Samir invested $89,000 in an account paying an interest rate of 6¹/₄% compounded monthly.

To calculate the time it would take Samir's money to double, use the compound interest formula.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

As the principal amount is doubled, then A = 2P.

Given values:

Substitute the values into the formula and solve for t:

[tex]\implies 2P=P\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]

[tex]\implies 2=\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]

[tex]\implies 2=\left(1+0.005208333...\right)^{12t}[/tex]

[tex]\implies 2=\left(1.005208333...\right)^{12t}[/tex]

[tex]\implies \ln 2=\ln \left(1.005208333...\right)^{12t}[/tex]

[tex]\implies \ln 2=12t \ln \left(1.005208333...\right)[/tex]

[tex]\implies t=\dfrac{\ln 2}{12 \ln \left(1.005208333...\right)}[/tex]

[tex]\implies t=11.1192110...[/tex]

Therefore, it will take 11.12 years for Samir's investment to double.

[tex]\hrulefill[/tex]

To calculate how much longer it would take for Samir's money to double than for Penelope's money to double, subtract the value of t for Penelope from the value of t for Samir:

[tex]\begin{aligned}\implies t_{\sf Samir}-t_{\sf Penelope}&=11.1192110......-10.872896949...\\&= 0.246314066...\\&=0.25\; \sf years\;(nearest\;hundredth)\end{aligned}[/tex]

Therefore, it would take 0.25 years longer for Samir's money to double than for Penelope's money to double.

The critical value t Subscript c for a confidence level of 0.90 and sample size of 26 is 1.708. A t-value greater than or less than 1.708 in absolute value would lead to rejection of the null hypothesis at the 0.10 level of significance.

To find the critical value t Subscript c for the confidence level c=0.90 and sample size n=26, we can use a t-distribution table or calculator.

Since we have a sample size of n=26, we have n-1 = 25 degrees of freedom. Using a t-distribution table or calculator with 25 degrees of freedom and a confidence level of 0.90, we get

t Subscript c = 1.708

Therefore, the critical value t Subscript c for the confidence level c=0.90 and sample size n=26 is 1.708. This means that if we calculate the t-value from our sample data and it is greater than or less than 1.708 in absolute value, we can reject the null hypothesis at the 0.10 level of significance.

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If the probability that this teacher is female is 3/5 , there are a total of 90 teachers at the school.

Let's denote the total number of teachers at the school as T. We know that the probability of choosing a female teacher is 3/5. Therefore, the probability of choosing a male teacher is 1 - 3/5 = 2/5.

We are also given that there are 36 male teachers at the school. We can use this information to set up an equation:

36/T = 2/5

To solve for T, we can cross-multiply:

36 x 5 = 2 x T

180 = 2T

T = 90

Therefore, there are a total of 90 teachers at the school.

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Complete question is:

One of the teachers at a school is chosen at random. The probability that this teacher is female is 3/5 There are 36 male teachers at the school. Work out the total number of teachers at the school.

Answer

Consider the time taken to completion time (in months) for the construction of a particular model of homes: 4.1 3.2 2.8 2.6 3.7 3.1 9.4 2.5 3.5 3.8 Find the mean, median mode, first quartile and third quartile. Find the outlier?

To find the mean, we add up all the values and divide by the number of values:

Mean = (4.1 + 3.2 + 2.8 + 2.6 + 3.7 + 3.1 + 9.4 + 2.5 + 3.5 + 3.8) / 10

Mean = 36.7 / 10

Mean = 3.67

To find the median, we need to put the values in order:

2.5, 2.6, 2.8, 3.1, 3.2, 3.5, 3.7, 3.8, 4.1, 9.4

The middle number is the median, which is 3.35 in this case.

To find the mode, we look for the value that appears most often. In this case, there is no mode as no value appears more than once.

To find the first quartile (Q1), we need to find the value that separates the bottom 25% of the data from the top 75%. We can do this by finding the median of the lower half of the data:

2.5, 2.6, 2.8, 3.1, 3.2

The median of this lower half is 2.8, so Q1 = 2.8.

To find the third quartile (Q3), we need to find the value that separates the bottom 75% of the data from the top 25%. We can do this by finding the median of the upper half of the data:

3.7, 3.8, 4.1, 9.4

The median of this upper half is 3.95, so Q3 = 3.95.

To find the outlier, we can use the rule that any value more than 1.5 times the interquartile range (IQR) away from the nearest quartile is considered an outlier. The IQR is the difference between Q3 and Q1:

IQR = Q3 - Q1

IQR = 3.95 - 2.8

IQR = 1.15

1.5 times the IQR is 1.5 * 1.15 = 1.725.

The only value that is more than 1.725 away from either Q1 or Q3 is 9.4. Therefore, 9.4 is the outlier in this data set.

Answer

To find the perimeter of a regular polygon with n sides, we can use the formula:

Perimeter = n * s

where s is the length of each side. To find s, we can use trigonometry to find the length of one of the sides and then multiply by the number of sides.

In a regular polygon with n sides, the interior angle at each vertex is given by:

Interior angle = (n - 2) * 180 degrees / n

In a 15-sided polygon, the interior angle at each vertex is:

(15 - 2) * 180 degrees / 15 = 156 degrees

If we draw a line from the center of the polygon to a vertex, we form a right triangle with the side of the polygon as the hypotenuse, the distance from the center to the vertex as one leg, and half of the side length as the other leg. Using trigonometry, we can find the length of half of the side:

sin(78 degrees) = 12 / (1/2 * s)

s = 2 * 12 / sin(78 degrees)

s ≈ 2.17 inches

Finally, we can find the perimeter of the polygon:

Perimeter = 15 * s

Perimeter ≈ 32.55 inches

Rounding this to the nearest whole number, we get that the best approximation for the perimeter is 33 inches. Therefore, the closest option is (1) 68 inches.

The equation of the ellipse is x^2/9 + y^2/6.75 = 1

To find the equation of an ellipse, we need to know the center, the major and minor axis, and the foci.

Since we are given the eccentricity and foci, we can use the following formula:

c = (1/2)a

Since the foci are (0, +/-3), the center is at (0, 0). We know that c = 3/2, so we can find a:

c = (1/2)a

3/2 = (1/2)a

a = 3

The distance from the center to the end of the minor axis is b, which can be found using the formula:

b = √(a^2 - c^2)

b = √(3^2 - (3/2)^2)

b = √6.75

So the equation of the ellipse is:

x^2/a^2 + y^2/b^2 = 1

Plugging in the values we found, we get:

x^2/3^2 + y^2/6.75 = 1

Simplifying:

x^2/9 + y^2/6.75 = 1

Therefore, the equation of the ellipse is x^2/9 + y^2/6.75 = 1

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At the point P(2,2), the unit vector for the direction of steepest ascent is  (-i + j)/√2, and the unit vector for the direction of steepest descent is (i - j)/√2. A vector that points in the direction of no change in the function at P is (2e^(-1/3)/49) i + (2e^(-1/3)/49) j + (2/7) k.

To find the unit vectors that give the direction of steepest ascent and steepest descent at P, we need to find the gradient of F at P and normalize it to obtain a unit vector.

First, we find the partial derivatives of F with respect to x and y

Fx = -2x e^(-x^2/(16-y^2))/((16-y^2)^2)

Fy = 2y e^(-x^2/(16-y^2))/((16-y^2)^2)

Plugging in the coordinates of P, we get

Fx(2,2) = -2e^(-1/3)/49

Fy(2,2) = 2e^(-1/3)/49

Therefore, the gradient of F at P is

∇F(2,2) = (-2e^(-1/3)/49) i + (2e^(-1/3)/49) j

To obtain the unit vector in the direction of steepest ascent, we normalize the gradient

u = (∇F(2,2))/||∇F(2,2)|| = (-i + j)/√2

To obtain the unit vector in the direction of steepest descent, we take the negative of u

v = -u = (i - j)/√2

To find a vector that points in a direction of no change in the function at P, we need to find a vector orthogonal to the gradient of F at P. One way to do this is to take the cross product of the gradient with the vector k in the z-direction

w = ∇F(2,2) x k = (2e^(-1/3)/49) i + (2e^(-1/3)/49) j + (2/7) k

Therefore, the vector that points in a direction of no change in the function at P is

(2e^(-1/3)/49) i + (2e^(-1/3)/49) j + (2/7) k

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Answer:

To find (f + g)(x), we need to add the two functions f(x) and g(x), and then evaluate the sum at x.

So, we have:

(f + g)(x) = f(x) + g(x)

Substituting the given functions, we get:

(f + g)(x) = (2x + 3) + (3x + 2)

Simplifying the expression, we get:

(f + g)(x) = 5x + 5

Therefore, (f + g)(x) is equal to 5x + 5.