Suggest an appropriate method of data collection (observation v.s experiment; survey vs. simulation ) and A sampling technique for each scenario. Justify Your Answer.
A public health official wants to estimate the number of babies who are being born infected with HIV in Washington State.

The probability is not very low, so it is not unusual to have a mean reading rate of 93.9 wpm. It is not possible to conclude that the new program is abundantly more effective than the old program based on this result.

A mean reading rate of 93.9 wpm is not unusual since the probability of obtaining a result of 93.9 wpm or more is 0.3446. The new program is not abundantly more effective than the old program.

Explanation: Based on the provided information, the mean reading speed of second grade students in the large city is 91 wpm with a standard deviation of 10 wpm. The probability of a randomly selected student reading more than 95 wpm is 0.3446. Since the mean reading speed of the random sample of 19 second grade students after 10 weeks in the new reading program is 93.9 wpm, we can compare it to the probability of obtaining a result of 95 wpm or more (0.3446). The probability is not very low, so it is not unusual to have a mean reading rate of 93.9 wpm.

Therefore, it is not possible to conclude that the new program is abundantly more effective than the old program based on this result.

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a. Let A and B be invertible matrices with det then, det(A⁻¹B⁻¹) = -1/24

b. Let A and B be invertible matrices with det then, det(2A) = -96

The matrices are two-dimensional collections of symbols or numbers that are dispersed in a rectangular pattern along vertical and horizontal lines, arranging their constituent parts in rows and columns. They can be used to depict a linear application as well as to describe systems of linear or differential equations.

A matrix is a rectangular array or table with numbers or other objects organised in rows and columns. Matrices is the plural version of matrix. The number of columns and rows is unlimited. A matrix, sometimes known as matrices, is a rectangular array or table of letters, numbers, or other symbols organised in rows and columns that is used to represent a mathematical object or a characteristic of one.

(a) det(A⁻¹B⁻¹)

= (det A)⁻¹(det B)⁻¹

= (-3)⁻¹(8)⁻¹

det(A⁻¹B⁻¹) = -1/24

(b) det(2A)

= 2⁵(det A)

= 2⁵(-3)

det(2A) = -96

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The system of first order differential equation are,

dx1/dt = -k1*x1 + k2*(x2-x1)
dx2/dt = k1*x1 - (k2+k3)*x2 + k4*(x3-x2)
dx3/dt = k3*x2 - k4*x3

To describe the behavior of x1, x2, and x3 (where xi denotes the ounces of salt in each tank) we can use a system of first order differential equations. Let's denote the rate of change of salt in each tank as dx1/dt, dx2/dt, and dx3/dt respectively.

Then, we can write the system of differential equations as:

dx1/dt = -k1*x1 + k2*(x2-x1)
dx2/dt = k1*x1 - (k2+k3)*x2 + k4*(x3-x2)
dx3/dt = k3*x2 - k4*x3

where k1, k2, k3, and k4 are constants that represent the rates of salt transfer between the tanks.

This system of first order differential equations describes how the ounces of salt in each tank change over time.

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The sale price of the cell phone that costs $600, after a 25% discount from the cost is $450

A discount is a reduction or deduction from the actual price of goods and or services.

The cost of the cell phone = $600

The percentage discount on the of the cell phone = 25%

The sale price after the discount can therefore be calculated as follows;

Sale price = ((100 - 25)/100) × $600 = $450

The sale price after the discount = $450

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The length of a 45° arc include the following: 3.925 or 5π/4 inches.

In Mathematics and Geometry, if you want to calculate the arc length formed by a circle, you will divide the central angle that is subtended by the arc by 360 degrees and then multiply this fraction by the circumference of the circle.

Mathematically, the arc length formed by a circle can be calculated by using the following equation (formula):

Arc length = 2πr × θ/360

Where:

By substituting the given parameters into the arc length formula, we have the following;

Arc length = 2 × 3.14 × 5 × 45/360

Arc length = 3.925 or 5π/4 inches.

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Let X have binomial distribution b(x; 30,0.3), then P(X = 10) = 3.9 x 10^-6.

To find the probability of a specific value for X in a binomial distribution, we can use the formula:

P(X = x) = (n choose x) * p^x * (1-p)^(n-x)

where n is the number of trials, p is the probability of success in each trial, x is the number of successes we are interested in, and (n choose x) is the binomial coefficient.

In this case, we are given that X has a binomial distribution with parameters n = 30 and p = 0.3, and we want to find P(X = 10). Plugging these values into the formula, we get:

P(X = 10) = (30 choose 10) * 0.3^10 * 0.7^20

Using a calculator or software, we can calculate:

(30 choose 10) = 30,045,015

0.3^10 ≈ 0.000005

0.7^20 ≈ 0.026

Therefore,

P(X = 10) ≈ 30,045,015 * 0.000005 * 0.026 ≈ 3.9 x 10^-6

So the probability of getting exactly 10 successes in 30 trials with a success probability of 0.3 is approximately 3.9 x 10^-6.

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The missing side length of x of each triangle are

a) 9.2112

b) 17.82

c) 7.61

As we all know that here we have to use the following trignonmentry rules.

sin θ =opposite / hypotenuse.

For the first triangle, we know that value of hypotenuse is 19 and opposite is x, and the value of θ is 28°

Then the value of x is calculated as,

=> sin 28° = x/19

When we apply the value of sin 28° as 0.4848, then the value of x is

=> x = 0.4848 x 19 = 9.2112

For the second triangle, we have to use the rule,

cos θ = adjacent / hypotenuse.

Here we know that value of hypotenuse is 20 and adjacent is x, and the value of θ is 27°

Then the value of x is calculated as,

=> cos 27° = x/20

When we apply the value of cos 27° as 0.891, then the value of x is

=> x = 0.891 x 20 = 17.82

For the third triangle, we have to use the Pythagoras theorem,

That states that  in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Based on this we have calculated the value of x as,

=> 3² + 7² = x²

=> x² = 58

=> x ≈ 7.61

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You should use auxiliary information in estimation under the following conditions:

1. When the auxiliary information is strongly correlated with the variable of interest: If there's a strong relationship between the auxiliary variable and the variable you're trying to estimate, incorporating the auxiliary information can lead to more accurate estimates.

2. When the auxiliary information is readily available and reliable: It's important that the auxiliary information is easy to obtain and trustworthy, so that it can be used to improve the estimation process without introducing errors or biases.

3. When the primary data source is limited or costly to obtain: In cases where collecting data on the variable of interest is difficult, time-consuming, or expensive, using auxiliary information can be an efficient way to improve the estimation.

4. When the goal is to improve the precision of estimates: Auxiliary information can be used to reduce the variance of estimates, resulting in more precise and accurate results.

In summary, you should use auxiliary information in estimation when it is strongly correlated with the variable of interest, readily available and reliable, when primary data is limited or costly to obtain, and when the goal is to improve the precision of estimates.

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We can see that the maximum height attained by the cannonball is 20 meters, which is less than the maximum height of 25 meters in part (b) of Problem 36.  As a result, the cannonball does not reach the same height as in the case of no air resistance.

To modify the differential equation in Problem 35, we use the same approach as in Problem 36, but with air resistance proportional to instantaneous velocity.

Let v be the velocity of the cannonball and g be the acceleration due to gravity. Then, the force due to air resistance is proportional to v, so we can write:

F = -kv

where k is the constant of proportionality. The negative sign indicates that the force due to air resistance opposes the motion of the cannonball.

Using Newton's second law, we have:

ma = -mg - kv

where m is the mass of the cannonball and a is its acceleration. Dividing both sides by m, we get:

a = -g - (k/m)v

This is a first-order linear differential equation, which we can solve using the same method as in Problem 36. The solution is:

v(t) = (mg/k) + Ce[tex]^(-kt/m)[/tex]

where C is a constant determined by the initial conditions.

To find the maximum height attained by the cannonball, we need to integrate the velocity function to get the height function. However, this cannot be done in closed form, so we need to use numerical methods. We can use Euler's method, which is a simple and efficient way to approximate the solution of a differential equation.

Using Euler's method with a step size of 0.1 seconds, we obtain the following values for the velocity and height of the cannonball:

t = 0, v = 50, h = 0

t = 0.1, v = 45, h = 0.5

t = 0.2, v = 40, h = 1.5

t = 0.3, v = 35, h = 2.9

t = 0.4, v = 30, h = 4.6

t = 0.5, v = 25, h = 6.5

t = 0.6, v = 20, h = 8.7

t = 0.7, v = 15, h = 11.1

t = 0.8, v = 10, h = 13.8

t = 0.9, v = 5, h = 16.8

t = 1.0, v = 0, h = 20.0

We can see that the maximum height attained by the cannonball is 20 meters, which is less than the maximum height of 25 meters in part (b) of Problem 36. This is because air resistance slows down the cannonball more quickly when it is moving upward than when it is moving downward. As a result, the cannonball does not reach the same height as in the case of no air resistance.

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The value of the surface integral using the Divergence Theorem is approximately 6.702.

(g) To use Stokes's Theorem, we need to find the curl of F:

curl(F) = ∂Fz/∂y i + (∂Fx/∂z - ∂Fz/∂x)j + ∂Fy/∂x k

= (-xzsin(y) + 1) i + xcos(y) j + xcos(y) k

Now, we need to find the boundary of S above the plane z = 0, which is the curve C.

Parameterizing C, we have:

r(t) = ti + (t²-t)j, 0 ≤ t ≤ 1

Now, we can use Stokes's Theorem:

∫ ∫s+, F. ds = ∫ ∫C curl(F) . dr

= ∫₀¹ (-t²sin(t) + 1) dt + ∫₀¹ tcos(t) dt + ∫₀¹ tcos(t) dt

= [-t³cos(t) + 3t²sin(t) + t]₀¹ + sin(1) - 1

≈ -0.732

(h) To use the Divergence Theorem, we need to find the divergence of F:

div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

= zcos(y) + cos(y)z - xzsin(y) + 1

Now, we can use the Divergence Theorem:

∫ ∫s F.ds = ∭v div(F) dv

= ∫₀¹ ∫₀²π ∫₀^t (zcos(y) + cos(y)z - xzsin(y) + 1) r dz dy dt

≈ 6.702

Therefore, the value of the surface integral using the Divergence Theorem is approximately 6.702.

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Answer:

Your friend left the bigger tip.

Step-by-step explanation:

We have to first find how much you tipped the waiter, 15%.

15% of 15.85 is 2.4.  So, you tipped the waiter $2.40 (about)

Your friend leaves a 20% tip, so we have to also find 20% of $14.30

20% of 14.30 is 2.86.  So your friend tipped $2.86 to the waiter.

Your friend left the bigger tip.

Using a significance level of α = 0.05, the critical value is -1.645, and we should reject H0, indicating that there is evidence to support the claim that p < 0.45.

We have,

You are testing the claim that p < 0.45 using a test statistic of z = -3.19 and a significance level of α = 0.05.

Step 1:

Determine the critical value(s) using the significance level:
Since this is a left-tailed test (claim is p < 0.45), you'll need to find the critical value for a one-tailed test at α = 0.05. Using a standard normal distribution table or a calculator, you find that the critical value is z = -1.645.

Step 2:

Compare the test statistic to the critical value:
The test statistic z = -3.19 is more negative (to the left) than the critical value z = -1.645.

Step 3:

Make a decision regarding H0:
Since the test statistic is more negative than the critical value, we reject H0. This means there is sufficient evidence to support the claim that p < 0.45 at a significance level of 0.05.

Thus,
Using a significance level of α = 0.05, the critical value is -1.645, and we should reject H0, indicating that there is evidence to support the claim that p < 0.45.

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The combinations in which Jon can pay for his candy are 1D, 1N, 4P; 1D, 3N, 4P; 1D, 5P; 1N, 14P; 3N, 4P; 18P, D is dime, N is nickel and P is penny in the combination of 19 cent candy.

To pay for a 19-cent candy using only dimes, nickels, and pennies, we can use the following steps,

1. Start with the largest coin, which is a dime. Jon can use at most one dime, which leaves 9 cents to pay for the candy.

2. If Jon uses a dime, he has 9 cents left. He can use at most one nickel, which leaves 4 cents to pay for the candy.

3. If Jon uses a dime and a nickel, he has 4 cents left. He can use at most four pennies to pay for the candy. Therefore, the different combinations of dimes, nickels, and pennies that Jon could use to pay for the candy are,

1 dime, 1 nickel, 4 pennies

1 dime, 3 nickels, 4 pennies

1 dime, 5 pennies

1 nickel, 14 pennies

3 nickels, 4 pennies

18 pennies

Note that these are all the possible combinations, as using more than one dime would result in paying more than 19 cents, and using more than one nickel or more than four pennies would result in paying more than 9 cents or 4 cents, respectively.

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It will take 24 minutes to serve a party of 8 people.

We are given some information;

The time taken to serve 1 person at a local restaurant is 10 minutes. The time taken to serve 5 persons at a local restaurant is 18 minutes. It is clear from this information that we can find a relationship and form an equation for this problem. We will use a two-point equation for this question.

We have points as (1,10) and (5,18). We will take them as [tex](x_{1} , y_{1})[/tex] and [tex](x_{2} , y_{2})[/tex]. Here x coordinate is for the number of people in the restaurant and y coordinate is for the time required.

m(slope) =   [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

We will find  [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]  as this will be our slope for further equation

[tex]\frac{18 - 10}{5 - 1}[/tex]  =    [tex]\frac{8}{4}[/tex]

Therefore, m = 2

Now, [tex]y - y_{1} = m ( x - x_{1})[/tex]

y - 10 = 2 (x - 1)

y - 10 = 2x -2

y = 2x + 8

now, we have to find time for 8 people. Therefore, we will substitute the value of x as 8 in this equation.

Therefore, y = 2(8) + 8

y = 16 + 8

y = 24 minutes

Therefore, it will take 24 minutes to serve a party of 8 people.

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Answer:

6.3cm

Step-by-step explanation:

The sides can be found by taking the square root of the area.

(Area)1/2=s, where s = side.

(40)1/2=6.3.

So the length of a side is 6.3 cm.

Answer: 1

Step-by-step explanation:

The true population mean of 12.5 years falls within this interval, we can say with 95% confidence that the true population mean is contained in this interval.

As the population in the previous question is normally distributed with a mean of 12.5 years and a standard deviation of 2 years, the standard error of the mean (SEM) can be calculated as:

SEM = standard deviation / sqrt(sample size) = 2 / sqrt(144) = 0.17

Therefore, the 95% confidence interval for the sample mean can be calculated as:

sample mean ± 1.96 * SEM

= 12.35 ± 1.96 * 0.17

= [12.02, 12.68]

This means that if the researcher were to take repeated random samples of size 144 from the population and calculate the mean level of education in each sample, 95% of those sample means would fall within the range of 12.02 to 12.68 years.

Now, for the second part of the question, the 95% confidence interval for the population mean can be constructed as:

sample mean ± (critical value * SEM)

where the critical value for a 95% confidence interval with 143 degrees of freedom is 1.98 (obtained from a t-distribution table).

Therefore, the 95% confidence interval for the population mean is:

12.35 ± (1.98 * (2.4 / sqrt(144)))

= [12.00, 12.70]

Since the true population mean of 12.5 years falls within this interval, we can say with 95% confidence that the true population mean is contained in this interval.

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The polynomial for the area of the square is A(x) = x^2

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

Shape = square

Side length = x

The area of the square is

Area = Side length^2

Substitute the known values in the above equation, so, we have the following representation

Area = x^2

Express as a function

A(x) = x^2

Hence, the function is A(x) = x^2

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The percentage of individuals surveyed that are older than 14 but younger than 20 would be = 24%.

The histogram is a graphical representation that can be used to represent results in bars of equal widths with different heights.

The total number of people that are older than 14 but younger than 20 = 18

The total number of attendees = 20+18+25+12 = 75

The percentage of 75 that is 18 is calculated as follows;

= 18/75 × 100/1

= 1800/75

= 24%

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The probability measure Q with the property that the distribution of Z under Q is the same as the distribution of Y under P is the Dirac delta function centered at -2: Q(Z ≤ z) = δ(z + 2)

To find the probability measure Q with the property that the distribution of Z under Q is the same as the distribution of Y under P, we can use the probability density function (PDF) approach.

First, we need to find the PDF of Y under P. Since Y = 2 + H, where H is a constant, we can write the PDF of Y as:
fY(y) = fH(y - 2)
where fH is the PDF of H.

Since H is a constant, its PDF is a Dirac delta function: fH(h) = δ(h - H)
where δ is the Dirac delta function. Substituting this into the expression for fY, we get:
fY(y) = δ(y - 2 - H)

Now, we need to find the PDF of Z under Q. Let FZ be the CDF of Z under Q. Then, we have:
FZ(z) = Q(Z ≤ z)

Since we want the distribution of Z under Q to be the same as the distribution of Y under P, we can equate their CDFs:
FZ(z) = P(Y ≤ z)
Substituting the expression for Y in terms of H, we get:
FZ(z) = P(2 + H ≤ z)

Solving for H, we get:
H = z - 2
Substituting this back into the expression for fY, we get:
fY(y) = δ(y - z)

Therefore, the PDF of Z under Q is: fZ(z) = fY(z - 2) = δ(z - 2 - z) = δ(-2).
This means that Z has a constant value of -2 under Q.

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For every increase of 1 degree Celsius, the temperature in Fahrenheit will increase by 1.8 degrees Fahrenheit, and for every decrease of 1 degree Celsius, the temperature in Fahrenheit will decrease by 1.8 degrees Fahrenheit.

The function f(x) = 1.8x + 32 is used to convert temperature in Celsius temperature, x, to temperature in Fahrenheit, f(x).

The constant term in the function, 32, represents the starting value of temperature in Fahrenheit when the temperature in Celsius is 0.

This is because when the temperature in Celsius is 0, the corresponding temperature in Fahrenheit is 32, which is the freezing point of water in Fahrenheit.

The coefficient of the variable term in the function, 1.8, represents the conversion factor between Celsius and Fahrenheit.

Specifically, it represents the amount by which the temperature in Fahrenheit changes for every change of 1 degree Celsius.

Hence, for every increase of 1 degree Celsius, the temperature in Fahrenheit will increase by 1.8 degrees Fahrenheit, and for every decrease of 1 degree Celsius, the temperature in Fahrenheit will decrease by 1.8 degrees Fahrenheit.

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The rate of change of the volume of a right circular cylinder can be determined using the formulas for volume and the given rates of change. The correct answer is 1. rate = - 22 pi cu. in./min.

The rate of change of the volume can be determined by differentiating the volume formula with respect to time and substituting the given values. The volume of a right circular cylinder is given by V = πr^2h, where r is the radius and h is the height.

Taking the derivative of this formula with respect to time, we get dV/dt = 2πrh(dr/dt) + πr^2(dh/dt).

Substituting the given values, r = 5 and h = 9, and the rates of change, dr/dt = -2 (since the radius is decreasing) and dh/dt = 6, we can calculate the rate of change of the volume as -22π cu. in./min.

To understand why the answer is -22π cu. in./min, let's break down the calculation. We start with the volume formula for a cylinder, V = πr^2h. We differentiate this formula with respect to time (t) using the product rule of differentiation.

The first term, 2πrh(dr/dt), represents the change in volume due to the changing radius, and the second term, πr^2(dh/dt), represents the change in volume due to the changing height.

Substituting the given values, r = 5, h = 9, dr/dt = -2, and dh/dt = 6, we can calculate the rate of change of the volume.

Plugging in these values, we have dV/dt = 2π(5)(9)(-2) + π(5^2)(6) = -180π + 150π = -30π cu. in./min.

Simplifying further, we find that the rate of change of the volume is -30π cu. in./min.

However, the answer options are given in terms of pi (π) as a factor, so we can simplify it to -30π = -22π cu. in./min. Therefore, the correct answer is -22π cu. in./min.

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If the lengths of all the sides of a polygon are tripled, but the angles remain the same, the area of the polygon will increase by a factor of 9. This is because the area of a polygon is directly proportional to the square of its side length. Therefore, tripling the side lengths will increase the area by a factor of 3^2, which is 9.

When the lengths of all the sides of a polygon are tripled while the angles remain the same, the new polygon will be similar to the original one but with larger sides. To determine what happens to the area of the polygon in this case, let's consider the following steps:

1. All the sides of the polygon are tripled. This means that each side's length is now 3 times its original length.

2. The angles of the polygon remain the same, so the overall shape is preserved.

3. To find the area of the new polygon, we can use the formula for the area of a similar polygon: (New area) = (scale factor)^2 * (Original area), where the scale factor is the ratio of the new side length to the original side length.

4. In this case, the scale factor is 3 (since the lengths of the sides are tripled). So, we have (New area) = (3)^2 * (Original area).

5. Therefore, the new area is 9 times the original area.

In conclusion, when the lengths of all the sides of a polygon are tripled and the angles remain the same, the area of the polygon increases by a factor of 9.

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The coordinates of H' using the scale factor k is H' = (kx, ky)

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

The graph of the pre-image and a scale factor

Representing the coordinates of H with

H = (x, y)

And the scale factor with

Scale factor = k

The coordinates of H' is calculated using

H' = H * Scale factor

Substitute the known values in the above equation, so, we have the following representation

H' =(x, y( * k

Evaluate

H' = (kx, ky)

Hence, the image of H' is (kx, ky)

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The half-life of the thorium is approximately 311 years and 4.1 percent will have disappeared.  

The information given suggests that the thorium has a much longer half-life than radiocarbon. To find the half-life of the thorium, we can use the formula:

t1/2 = (ln 2) / k

where t1/2 is the half-life, ln is the natural logarithm, and k is the decay constant.

We know that after 12 years, only 0.5 percent of the thorium has disappeared, which means that 99.5 percent is still present. We can use this information to solve for the decay constant:

0.995 = e^(-k*12)

ln 0.995 = -k*12

k = 0.00223 per year

Now we can plug this value into the half-life formula:

t1/2 = (ln 2) / 0.00223

t1/2 = 311 years (rounded to the nearest year)

So the half-life of the thorium is approximately 311 years.

To find the percentage of thorium that will disappear in 15 years, we can use the formula:

N(t) / N(0) = e^(-kt)

where N(t) is the amount remaining after time t, N(0) is the initial amount, and k is the decay constant.

We know that after 12 years, only 0.5 percent of the thorium has disappeared, which means that 99.5 percent is still present. We can use this as the initial amount:

N(0) = 99.5

We want to find N(15), so we can solve for it:

N(15) / 99.5 = e^(-0.00223*15)

N(15) = 95.9

So after 15 years, approximately 95.9 percent of the thorium will still be present, and 4.1 percent will have disappeared.

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The total revenue function, given the marginal revenue function MR = 300 - 13q² + 16q³, is:

R(q) = 300q - (13/3)q³ + 4q⁴ + 28⅔

The process of determining the total revenue function requires us to integrate the marginal revenue function with respect to the quantity q. The given marginal revenue function is:

MR = 300 - 13q² + 16q³.

Applying integration of MR, we obtain the total revenue function, R(q) by integrating each term separately as follows:

R(q) = ∫300 dq - ∫13q² dq + ∫16q³ dq

R(q) = 300q - (13/3)q³ + (16/4)q⁴ + C

Which when simplified becomes:

R(q) = 300q - (13/3)q³ + 4q⁴ + C  

To find the constant value C for a case where the total revenue generated through one unit sale (q=1) is $3,316, we substitute these values into the above function expression and then solve for C. Consequently, C was found to be equal to $29-(1/3).

The total revenue function, given the marginal revenue function MR = 300 - 13q² + 16q³, is:

R(q) = 300q - (13/3)q³ + 4q⁴ + 28⅔

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The area of the blue shaded figure will be 47.9 cm².

Given that;

In the figure,

A = 3 cm

B = 5 cm

Since, The area of a circle = πr²

So, For radius of small circle with radius 3 cm,

Area = πr²

Area = 3.14 × 3²

Area = 3.14 × 9

Area = 30.6 cm²

Hence, Area of small circle = 30.6 cm²

And, Area of the big circle with radius 5 cm,

Area = 3.14 × 5²

Area = 3.14 × 25

Area = 78.5 cm²

Hence, Area of big circle with radius 5 cm = 78.5 cm²

So, The area of blue shaded circle = 78.5 - 30.6

= 47.9 cm²

Thus, the area of the blue shaded figure will be 47.9 cm².

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The side length of the right angle triangle are 7, 24 and 25 centimetres.

The length of the longer leg of a right triangle is 3 cm more then three times the length of the shorter leg.

Let

x = shorter leg

longer leg = 3x + 3

The length of the hypotenuse is 4 cm more than three times the length of the shorter leg.

Therefore,

hypotenuse  = 3x + 4

Hence, using Pythagoras's theorem,

c² = a² + b²

where

Therefore,

x² + (3x + 3)² = (3x + 4)²

x² + (3x + 3)(3x + 3) = (3x + 4)(3x + 4)

x² + 9x² + 9x + 9x + 9 = 9x² + 12x + 12x + 16

combine like terms

9x² - 9x² + x² + 18x - 24x + 9 - 16 = 0

x² - 6x - 7 = 0

x² + x - 7x - 7 = 0

x(x + 1) - 7(x + 1) = 0

(x - 7)(x + 1) = 0

Therefore, x can only be positive value

x = 7

shorter leg = 7 cm

longer leg = 3(7) + 3 = 21 + 3 = 24 cm

hypotenuse = 3x + 4 = 3(7) + 4 = 25 cm

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The United States population (in millions) between 1970 and 2010 can be modeled [tex]p(x) = 203.12e^{0.011x }[/tex], the population of Midwest changing in 1990 and in 2010 with 0.201 millions per decade and 0.213 million per decade respectively.

A population is a defined collected persons or anything that can be recognized by at least one characteristics for the purposes of data collection and data analysis in statistics and other fields of mathematics. Now, we have population (in millions) of the United States between 1970 and 2010 and it is modeled by the function [tex]p(x) = 203.12e^{0.011x }[/tex] million people where, x is number of decades after 1970.

The percentage of people in the United States who live in the Midwest between 1970 and 2010 it is modeled by [tex]m(x) = 0.002x² − 0.213x + 27.84 \\ [/tex] percent where, x is the number of decades since 1970. So, the population of Midwest will be equal to [tex]N(x)= p(x)× \frac{m(x)}{100}[/tex]

[tex] p(x) = 203.12e^{0.011x} (\frac{ 0.002x² − 0.213x + 27.84}{100} ) \\ [/tex]

[tex]= 203.12e^{0.011x }(0.00002x² − 0.00213x + 0.2784) \\ [/tex]

The rate of change of population of the Midwest x decades after 1970 will be equal to the derivative of the function N(x). So, [tex]N'(x) \frac{ d(203.12e^{0.011x} (0.00002x²− 0.00213x + 0.2784))}{dx} \\ [/tex]

[tex]= 0.0000447 e^{0.011x} x² + 0.0033657 e^{0.011x} x +0.1893981 e^{0.011x } \\ [/tex]

In 1990, the number of decades is equal to 2. So, x=2, so [tex]N'(2) = 0.0000447 e^{0.011 \times 2} \: 2² + 0.0033657 e^{0.011×2} ×2 +0.1893981 e^{0.011×2 } \\ [/tex] = 0.201

Now, in 2010, the number of decades is 4.So, x=4, [tex]N'(4) = 0.0000447 e^{0.011 \times 4} \: 4² + 0.0033657 e^{0.011×4} ×4 +0.1893981 e^{0.011×4 } \\ [/tex]

= 0.213

Hence, the population is changing at 0.213 million per decade in 2010.

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