Mr. smith is taking an extravagant trip to Puerto Rico. He knows he can bring 1.5 bags for every $52.50 he pays. If his flight costs $262.50 how many bags can Mr. smith bring?

Answer:

I’m pretty sure it’s 6.4 feet

Step-by-step explanation:

Based on the diagram, we can see that we have a right triangle with the ladder, the wall, and the distance up the wall that the ladder reaches.

We know that the ladder is 8 feet long and makes a 53 degree angle with the wall. We can use trigonometry to find the height that the ladder reaches up the wall.

The trigonometric function that relates the angle, the opposite side, and the hypotenuse in a right triangle is the sine function.

In this case, the height up the wall is the opposite side and the ladder is the hypotenuse.

To calculate this value, we first need to find the sine of 53 degrees. The sine function is a trigonometric function that relates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. In this case, we want to find the sine of the angle that the ladder makes with the wall, which is 53 degrees.

The sine of an angle is calculated by dividing the length of the side opposite the angle by the length of the hypotenuse. In this case, the length of the side opposite the angle is the height up the wall that the ladder reaches, and the length of the hypotenuse is the length of the ladder, which is 8 feet.

So, we can use the sine function to find the height up the wall as follows:

sin(53) = opposite/hypotenuse

sin(53) = opposite/8

To isolate the value of "opposite" on one side of the equation, we can multiply both sides by 8:

8 * sin(53) = opposite

Now, we can substitute the value of sin(53), which is approximately 0.8, into the equation:

opposite = 8 * sin(53)

opposite = 8 * 0.8

opposite ≈ 6.4 feet

Therefore, the distance up the wall that the ladder reaches is closest to 6.4 feet.

The measure of the angle is <B is 110 degrees

It is important to note that supplementary angles are described as angles that sum up to 180 degrees.

From the information given, we have that;

m<A = x + 26

m>B = 2x + 22

Equate the angles

m<A +m<B = 180

x + 26 + 2x + 22 = 180

collect the like terms

3x = 180 - 48

3x = 132

x = 44

the measure of <B = 2x + 22 = 2(44) + 22 = 88 + 22 = 110 degrees

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Check the picture below.

so the horizontal lines are 4 and 12, and then we have a couple of slanted ones, say with a length of "c" each

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=\sqrt{a^2 + o^2} \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{4}\\ o=\stackrel{opposite}{3} \end{cases} \\\\\\ c=\sqrt{ 4^2 + 3^2}\implies c=\sqrt{ 16 + 9 } \implies c=\sqrt{ 25 }\implies c=5 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{\LARGE Perimeter}}{4+12+5+5}\implies \text{\LARGE 26}[/tex]

True. Regression is a functional relationship between two or more correlated variables, where one variable is used to predict another. This statistical method helps in understanding the relationship between variables and making predictions based on that information.

Regression analysis is a powerful tool in statistics that helps to identify the relationship between variables, and it can be used to make predictions or forecasts based on that relationship. It involves fitting a mathematical model to the data, and then using that model to estimate the value of one variable based on the values of the other variables. There are many different types of regression analysis, each suited to different types of data and research.

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μA and μB represent the mean number of red blood cells in the blood of infected mice treated with Drug A and Drug B, respectively.

Null hypothesis (H0): Drug A and Drug B have the same effect on decreasing the number of internal parasites in a population of cats, μA = μB.

Alternative hypothesis (Ha): Drug A is better than Drug B at decreasing the number of internal parasites in a population of cats, μA < μB.

μA and μB represent the mean number of internal parasites in the population of cats treated with Drug A and Drug B, respectively.

Null hypothesis (H0): University of Maryland students and University of Denmark students have the same success rate at shooting freethrows, pMD = pDK.

Alternative hypothesis (Ha): University of Maryland students are better than University of Denmark students at successfully shooting freethrows, pMD > pDK.

pMD and pDK represent the proportion of successful freethrows for University of Maryland and University of Denmark students, respectively.

Null hypothesis (H0): There is no difference between Drug A and Drug B in the number of red blood cells in the blood of infected mice, μA = μB.

Alternative hypothesis (Ha): There is a difference between Drug A and Drug B in the number of red blood cells in the blood of infected mice, μA ≠ μB.

μA and μB represent the mean number of red blood cells in the blood of infected mice treated with Drug A and Drug B, respectively.

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a)

The cost per payment is 155.5.

b)

The value of 3 in the standard deviation principle is 1376.711.

We have,

a)

To calculate E(X^ 105), we first need to find the probability distribution of X after applying the deductible of 105.

Any loss below 105 will result in no payment, and any loss above or equal to 105 will result in payment equal to the loss minus the deductible.

Thus, the probability distribution of the payments.

Payment: 0 0 0 29 29 75 75 75 75 105 245 419 419

Probability: 0 0 0 1/12 1/12 1/6 1/6 1/6 1/6 1/12 1/12 1/12 1/12

Using this probability distribution, we can calculate E(X^ 105) as follows:

E(X^ 105) = 0^2 * 0 + 29^2 * (1/12 + 1/12 + 1/12 + 1/12) + 75^2 * (1/6 + 1/6 + 1/6 + 1/6) + 105^2 * (1/12) + 245^2 * (1/12) + 419^2 * (1/12)

= 34390.5833

The cost per payment ex(105) is simply the expected payment per policyholder, which can be calculated as follows:

ex(105) = 29  (1/3) + 75 * (2/3) + 105 * (1/6) + 245 * (1/6) + 419 * (1/6)

= 155.5

b)

The standard deviation principle states that the total cost of claims, including the deductible, should be equal to the product of the standard deviation and the value of the insurance against risk.

In this case, the insurance against risk is the maximum amount that the insurance company is willing to pay per policyholder, which is 105. Thus, we have:

105 + Bo = s(X) x VaR

where s(X) is the standard deviation of X and VaR is the value at risk, which is the amount that the company expects to pay out with a certain probability (usually 99% or 99.5%).

We can solve for Bo as follows:

Bo = s(X) * VaR - 105

Assuming a VaR of 99%, we need to find the 1% percentile of X, which is the value x such that P(X ≤ x) = 0.01.

From the empirical distribution, we can see that the 1% percentile is 94. Thus, we have:

VaR = 105 - 94 = 11

To calculate s(X), we first need to find the mean of X, which is:

mean(X) = (94 + 3104 + 2134 + 4*180 + 210 + 350 + 524)/13 = 224

Using the formula for the sample standard deviation, we get:

s(X) = √((1/12)((94-224)^2 + 3(104-224)^2 + 2*(134-224)^2 + 4*(180-224)^2 + (210-224)^2 + (350-224)^2 + (524-224)^2))

= 142.701

Now,

Bo = 142.701 x 11 - 105

= 1376.711

Thus,

The cost per payment is 155.5.

The value of 3 in the standard deviation principle is 1376.711.

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Using the linear regression equation, we predict Mastura's revenue for the month to be approximately RM 1,165.55 when there are 49 departure delays.

The general form of a linear equation is:

Revenue = Intercept + (Coefficient for Delays * Number of Delays)

In this case, the Intercept is 729.48138, and the Coefficient for Delays is 8.9014135. So the equation becomes:

Revenue = 729.48138 + (8.9014135 * Number of Delays)

Now, we need to predict the revenue for the month when there are 49 departure delays:

Revenue = 729.48138 + (8.9014135 * 49)

Revenue = 729.48138 + (436.0690615)

Revenue = 1165.5504415

Thus, Mastura's revenue for the month is approximately RM 1,165.55 when there are 49 departure delays.

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The discriminant of f(x)is -28, and f(x) has no distinct real number zeros.

The expression[tex]b^{2}- 4ac[/tex] gives the value of discriminant of the quadratic function with the form f(x) = [tex]ax^{2} + bx + c[/tex]. This result is obtained through using this formula on the quadratic function, where f(x) = [tex]-4x^{2}+ 10x - 8[/tex]:  [tex]b^2 - 4ac = (10)^2 - 4(-4)(-8)[/tex] = 100-128 = -28. Hence, -28 is the discriminant of f(x).

The discriminant informs us of the characteristics of the quadratic equation's roots. There are two unique real roots if the discriminant index is positive. There is just one real root (with a multiplicity of 2) if the discriminator is zero. There are only two complicated roots (no real roots) if discrimination is negative.

Given that f(x)'s discriminant is minus (-28), we can conclude that there are no true roots. F(x) contains two complex roots as a result. This is further demonstrated by the fact that the parabola widens downward and does not cross the x-axis, as indicated by the fact that the coefficient of the [tex]x^{2}[/tex] term in f(x) is negative.

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The percentage of the day the student spend at school in the pie chart is derived to be 35 percentage.

In a pie chart, the size of each sector is proportional to the value it represents. Therefore, the percentage represented by each sector can be calculated by dividing the value of that sector by the total value and multiplying by 100.

We shall represent the percentage of the day the student spend at school with the letter x, such that:

x = (126 × 100)/360

x = 7 × 5

x = 35%

Therefore, the percentage of the day the student spend at school is derived to be 35 percentage.

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A researcher records the following scores on a working memory quiz for two samples. Sample B has the largest standard deviation.

To determine which sample has the largest standard deviation, we need to calculate the standard deviation for both samples. Here are the steps to calculate the standard deviation:

1. Find the mean (average) of each sample.
2. Calculate the difference between each score and the mean.
3. Square the differences.
4. Find the mean of the squared differences.
5. Take the square root of the mean of the squared differences.

Sample A:
1. Mean: (2+3+4+5+6+7+8)/7 = 5
2. Differences: -3, -2, -1, 0, 1, 2, 3
3. Squared differences: 9, 4, 1, 0, 1, 4, 9
4. Mean of squared differences: (9+4+1+0+1+4+9)/7 = 28/7 = 4
5. Standard deviation: √4 = 2

Sample B:
1. Mean: (4+5+6+7+8+9+10)/7 = 7
2. Differences: -3, -2, -1, 0, 1, 2, 3
3. Squared differences: 9, 4, 1, 0, 1, 4, 9
4. Mean of squared differences: (9+4+1+0+1+4+9)/7 = 28/7 = 4
5. Standard deviation: √4 = 2

Both samples have the same standard deviation of 2.

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The circumference of the cookies is 10π which is approximately 31.4 inches.

A circle is simply a closed 2-dimensional curved shape with no corners or edges.

The circumference of a circle is expressed mathematically as;

C = 2πr

Where r is radius and π is constant pi ( π = 3.14 )

Given tha, the radius of the cookies is 5 inches.

So, we can substitute this value into the formula and calculate the circumference:

C = 2πr

C = 2 × 3.14 × 5

C = 31.4 in

Therefore, the circumference is approximately 31.4 inches.

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The percentage of all possible values of the variable that are less than 6 is:

0.1587 * 100% = 15.87%

a. To find the percentage of all possible values of the variable that lie between 4 and 9, we need to find the z-scores corresponding to these values and then find the area under the normal curve between those z-scores.

The z-score for x = 4 is:

z = (4 - 8) / 2 = -2

The z-score for x = 9 is:

z = (9 - 8) / 2 = 0.5

Using a standard normal table or calculator, we find that the area to the left of z = -2 is 0.0228 and the area to the left of z = 0.5 is 0.6915. Therefore, the area between z = -2 and z = 0.5 is:

0.6915 - 0.0228 = 0.6687

So, the percentage of all possible values of the variable that lie between 4 and 9 is:

0.6687 * 100% = 66.87%

b. To find the percentage of all possible values of the variable that exceed 5, we need to find the area under the normal curve to the right of z = (5 - 8) / 2 = -1.5.

Using a standard normal table or calculator, we find that the area to the left of z = -1.5 is 0.0668. Therefore, the area to the right of z = -1.5 (and hence the percentage of all possible values of the variable that exceed 5) is:

1 - 0.0668 = 0.9332

So, the percentage of all possible values of the variable that exceed 5 is:

0.9332 * 100% = 93.32%

c. To find the percentage of all possible values of the variable that are less than 6, we need to find the area under the normal curve to the left of z = (6 - 8) / 2 = -1.

Using a standard normal table or calculator, we find that the area to the left of z = -1 is 0.1587. Therefore, the percentage of all possible values of the variable that are less than 6 is:

0.1587 * 100% = 15.87%

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(a) The equation of the least-squares regression line that describes the relationship between the distance traveled in miles (x) and the pounds of fuel consumed (y) is given by: y = -4702.64 + 21.282x
(b) If the plot shows a random scatter, it indicates that the linear regression model is appropriate.
(c) The y-intercept in the context of the problem is -4702.64, which represents the predicted pounds of fuel consumed when the distance traveled is zero miles.

(a) The equation of the least-squares regression line for predicting the pounds of fuel consumed based on the distance traveled in miles is:

Fuel Consumed (lbs) = -4702.64 + 21.282 Distance Travelled (miles)

where Fuel Consumed and Distance Travelled are the variables used in the equation.

(b) Based on the residual plot, it is appropriate to use the linear regression equation to make predictions. The plot shows that the residuals are randomly scattered around the horizontal line at zero, indicating that there is no pattern or trend in the residuals. This suggests that the linear regression model is a good fit for the data and that the assumptions of linearity and constant variance are not violated.

(c) The y-intercept (-4702.64) represents the estimated pounds of fuel consumed when the distance traveled is zero. However, this value is not statistically meaningful in the context of the problem, as passenger aircraft cannot consume fuel if they do not travel any distance. Therefore, the y-intercept should not be interpreted in this case. The p-value for the intercept is 0.022, which is less than 0.05, indicating that the y-intercept is statistically significant. However, it may not be practically meaningful or interpretable in this context.

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The N electron configuration for nitrogen is 1s22s22p3.

To provide a complete electron configuration for nitrogen using the terms "electron" and "orbital," follow these steps:

1. Determine the atomic number of nitrogen (N). The atomic number of nitrogen is 7, meaning it has 7 electrons.
2. Fill the orbitals in order of increasing energy. The order is 1s, 2s, 2p, 3s, 3p, and so on.Following these steps, the electron configuration for nitrogen is 1s^22s^22p^3. This means there are 2 electrons in the 1s orbital, 2 electrons in the 2s orbital, and 3 electrons in the 2p orbital, which sums up to 7 electrons, corresponding to the atomic number of nitrogen.In writing the electron configuration for nitrogen, the first two electrons will go in the 1s orbital. Since 1s can only hold two electrons, the next two electrons for N go in the 2s orbital. The remaining three electrons will go into the 2p orbital.

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The person must leave the money in the bank for approximately 10.8 years (rounded to the nearest tenth of a year) for it to reach $2900 with 5.75% interest compounded monthly.

We can use the formula for compound interest to solve this problem:

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

where:

A is the amount of money after t years

P is the principal (the initial amount of money invested)

r is the annual interest rate (as a decimal)

n is the number of times the interest is compounded per year

t is the time in years

We want to find t, the time required for the investment to grow from $1000 to $2900. We know that P = 1000 and A = 2900. We also know that r = 0.0575 (5.75% as a decimal) and that the interest is compounded monthly, so n = 12.

Substituting these values into the formula, we get:

2900 = [tex]1000(1 + 0.0575/12)^(12t)[/tex]

Dividing both sides by 1000, we get:

2.9 = [tex](1 + 0.0575/12)^(12t)[/tex]

Taking the natural logarithm of both sides, we get:

[tex]ln(2.9) = 12t ln(1 + 0.0575/12)[/tex]

Solving for t, we get:

[tex]t = ln(2.9) / (12 ln(1 + 0.0575/12))[/tex]

Using a calculator, we get:

t ≈ 10.8

Therefore, the person must leave the money in the bank for approximately 10.8 years (rounded to the nearest tenth of a year) for it to reach $2900 with 5.75% interest compounded monthly.

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Answer: blue

Step-by-step explanation:

blue

Answer:

Both are non-linear. The first equation is a rational function and the second is a quadratic.

Step-by-step explanation:

It may be worth investigating to see if there is an issue with the meter or billing.

Sample mean = 46.16, Sample standard deviation = 45.05

To find the z-score of a bill, we use the formula: z = (x - mean) / standard deviation

January: z = (14.1 - 46.16) / 45.05 = -0.71

May: z = (15.84 - 46.16) / 45.05 = -0.65

November: z = (115.17 - 46.16) / 45.05 = 1.55

The z-score tells us how many standard deviations a bill is away from the mean. A negative z-score means the bill is below the mean, and a positive z-score means the bill is above the mean.

Analysis:

Based on the mean and standard deviation, a normal bill would be between 1.06 and 91.26.

The z-scores for January and May are both below -2/3, which indicates they are slightly lower than normal bills but not very unusual. The z-score for November is above 1, which indicates it is higher than normal bills and may be considered unusual.

There may be times when you would accept an unusual bill if there was a reasonable explanation, such as extreme weather conditions or a change in energy usage. However, if the bill is consistently unusual over time, it may be worth investigating to see if there is an issue with the meter or billing.

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There are three ways to show that two triangles are similar:

Angle-angle Theorem (AA): Two triangles are comparable if they have two pairs of congruent angles.

Side-side-side Theorem (SSS): If the ratios of two triangles' corresponding sides are identical, the triangles are comparable.

The side-angle-side theorem (SAS) states that if the ratios of two pairs of comparable sides of two triangles are identical and their angles are congruent, the triangles are similar.

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Answer:

D. shapes with a right angle

Step-by-step explanation:

All shapes, parallelogram, irregular hexagon and the right triangle have or are capable of having a right angle. None of the other answers make sense either.

Hope this helps :)

Answer:

D. shapes with a right angle

Step-by-step explanation:

The largest rectangle inscribed in a circle of radius 1 has an area of 2√2 units.

Draw the circle of radius 1 and sketch the rectangle inscribed in it. Let the length of the rectangle be 2x and the width be 2y. By symmetry, we know that the diagonals of the rectangle will pass through the center of the circle.

The length of the diagonal is equal to the diameter of the circle, which is 2. Using the Pythagorean theorem, we can write an equation relating the side lengths of the rectangle and the diameter of the circle

(2x)² + (2y)² = 2²

Simplifying, we get

4x² + 4y² = 4

Dividing both sides by 4, we get

x² + y² = 1

We want to maximize the area of the rectangle, which is given by A = 4xy.

We can use the equation from step 5 to solve for y in terms of x

y² = 1 - x²

y = √(1 - x²)

Substituting this into the area formula, we get

A = 4x*√(1 - x²)

To maximize this function, we can take the derivative with respect to x and set it equal to zero

dA/dx = 4(1 - x²)^(-1/2) - 4x²(1 - x²)^(-3/2) = 0

Solving for x, we get x = 1/√(2) or x = -1/√(2).

We can discard the negative solution since we are looking for a positive length.

Using x = 1/√(2), we can find the corresponding value of y

y = √(1 - x²) = √(1 - 1/2) = √(1/2)

Finally, we can calculate the area of the rectangle using these values

A = 4xy = 4(1/√(2))(√(1/2)) = 2(√(2))

Therefore, the area of the largest rectangle that can be inscribed in a circle of radius 1 is 2(√(2)).

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The solution involves setting up equations for the accumulated value of the two accounts and equating them at the end of the fourth year and the end of the eighth year. Solving for k gives k = 6.75.

Let the initial deposit made by the student be denoted by X.

After 4 years, the amount in the UENR Credit Union account is X(1+4i) = X(1+4*0.075) = X(1.3).

For the lecturer's account, we need to use the force of interest formula to calculate the accumulated amount after 4 years

A(4) = Xe^∫[0,4] 2t/t²+k dt = X[tex]e^{2ln(2k+16)-2ln(2k)}[/tex]/2

A(4) = X((2k+16)/2k[tex])^{1/2}[/tex]

After 8 years, both accounts earn the same amount of interest. Therefore, the amount in the UENR Credit Union account is X(1+8i) = X(1+8*0.075) = X(1.6).

And for the lecturer's account

A(8) = Xe^∫[0,8] 2t/t²+k dt = X[tex]e^{4ln(2k+32)-4ln(2k)}[/tex]/2

A(8) = X((2k+32)/2k)²

Since the earned is the same for both accounts, we have

X(1.6) - X(1.3) = X((2k+32)/2k)² - X((2k+16)/2k[tex])^{1/2}[/tex]

Simplifying the above equation gives

0.3X = X[(2k+32)/2k)² - (2k+16)/2k[tex])^{1/2}[/tex]]

Dividing both sides by X gives

0.3 = [(2k+32)/2k)² - ((2k+16)/2k[tex])^{1/2}[/tex]]

Squaring both sides and rearranging gives

16k³ - 60k² - 71k - 144 = 0

This cubic equation can be solved using numerical methods or by factoring it using trial and error. After solving, we get

k = 6.75 (approx)

Therefore, the value of k is approximately 6.75.

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Scatterplot would  more appropriate for displaying and describing the relationship between the age and the number of vocabulary words.

We have to compare scatterplot and line graph.

The association between age and vocabulary size would be better represented and explained by a scatter plot.

The link between two continuous variables is shown using a scatter plot; in this instance, age is a continuous variable whereas the quantity of vocabulary items is a discrete variable.

The relationship between the two variables can be visually examined with the use of scatter plots, which also reveal the direction and strength of the association.

Thus, the scatterplot is best choice.

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The equation of the graph in the dotted line is

The equation graphed on a dotted line is obtained from the knowledge of parabolic equation and transformation

From the parent function, which has the formula y = x^2, a reflection was noticed resulting to equation

y = -x^2

Then a translation to 3 units to the left, results to the equation of the form

The graph of the function is plotted  and attached

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The correct range of the given function is :

c. {y € R, y>=0}.

To determine the range of the function y = (x-2)^(1/2), we need to consider the possible values of y that can be obtained for different values of x.

a. {x € R} means that x can take any real value. However, since the square root of a negative number is not a real number, y can only take non-negative values. So, the range is {y € R, y>=0}.

b. {x € R, x>=2} means that x can take any real value greater than or equal to 2. Again, the square root of a negative number is not a real number, so y can only take non-negative values. So, the range is {y € R, y>=0}.

d. {y € R} means that y can take any real value. However, if we plug in a value of x less than 2, we get a negative value under the square root, which is not a real number. So, the range is not {y € R}.

Therefore, the correct answer is c. {y € R, y>=0}.

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the solution to the given differential equation is: Q(t) = -t^4/25 + t^4/5 ln(t) + C/t

To solve the given differential equation t dQ/dt + Q = t^4 ln(t), we'll first find the integrating factor, solve for Q(t), and then substitute the given terms.

Step 1: Find the integrating factor.
The integrating factor is e^(∫P(t)dt), where P(t) = 1/t in this case. So,

∫(1/t)dt = ln(t)

The integrating factor is e^(ln(t)) = t.

Step 2: Multiply the equation by the integrating factor.
t (t dQ/dt) + t(Q) = t^2 dQ/dt + tQ = t^5 ln(t)

Step 3: Integrate both sides of the equation.
∫(t^2 dQ/dt + tQ)dt = ∫(t^5 ln(t))dt

Using integration by parts on the right side (u = ln(t), dv = t^5 dt):

∫(t^5 ln(t))dt = (t^5 ln(t) / 5) - ∫(t^4 dt) = (t^5 ln(t) / 5) - (t^5 / 25) + C

Step 4: Solve for Q(t).
Since ∫(t^2 dQ/dt + tQ)dt = tQ, we have:

tQ = (t^5 ln(t) / 5) - (t^5 / 25) + C
Q(t) = -t^4/25 + t^4/5 ln(t) + C/t

So, the solution to the given differential equation is:

Q(t) = -t^4/25 + t^4/5 ln(t) + C/t

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z1 = (78.1 - 83) / (7/√49) ≈ -1.49 and z2 = (85.2 - 83) / (7/√49) ≈ 0.85. Using a standard normal distribution table or calculator, we find that P(-1.49 < z < 0.85) ≈ 0.6924. Therefore, P(78.1 < x < 85.2) ≈ 0.6924.

(a) Since the sample size is large enough (n ≥ 30) and the population standard deviation is known, the central limit theorem can be applied to conclude that the sampling distribution of the sample mean, x, is approximately normal with mean µ = 83 and standard deviation σ/√n = 7/√49 = 1.

(b) To find P(x > 84.9), we need to standardize the value of 84.9 using the formula z = (x - µ) / (σ/√n). Thus, z = (84.9 - 83) / (7/√49) = 1.9. Using a standard normal distribution table or calculator, we find that P(z > 1.9) ≈ 0.0287. Therefore, P(x > 84.9) ≈ 0.0287.

(c) To find P(x ≤ 76.7), we again need to standardize the value of 76.7 using the formula z = (x - µ) / (σ/√n). Thus, z = (76.7 - 83) / (7/√49) = -1.86. Using a standard normal distribution table or calculator, we find that P(z < -1.86) ≈ 0.0317. Therefore, P(x ≤ 76.7) ≈ 0.0317.

(d) To find P(78.1 < x < 85.2), we first standardize the values of 78.1 and 85.2 using the formula z = (x - µ) / (σ/√n). Thus, z1 = (78.1 - 83) / (7/√49) ≈ -1.49 and z2 = (85.2 - 83) / (7/√49) ≈ 0.85. Using a standard normal distribution table or calculator, we find that P(-1.49 < z < 0.85) ≈ 0.6924. Therefore, P(78.1 < x < 85.2) ≈ 0.6924.

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The Taylor series for f(x) centered at a = 5p is:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

To find the derivatives of f(x), we use the chain rule and the derivative of cos x:

f(x) = 4 cos x
f'(x) = -4 sin x
f''(x) = -4 cos x
f'''(x) = 4 sin x
f''''(x) = 4 cos x
...

Substituting a = 5p and evaluating the derivatives at a, we get:

f(5p) = 4 cos(5p) = 4
f'(5p) = -4 sin(5p) = 0
f''(5p) = -4 cos(5p) = -4
f'''(5p) = 4 sin(5p) = 0
f''''(5p) = 4 cos(5p) = 4
...

Therefore, the Taylor series for f(x) centered at a = 5p is:

f(x) = 4 - 4(x-5p)^2/2! + 4(x-5p)^4/4! - ...

Simplifying the series, we get:

f(x) = 4 - 2(x-5p)^2 + (x-5p)^4/3! - ...

Note that this is the Maclaurin series for cos x, with a = 0, multiplied by 4 and shifted to the right by 5p.

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