Question 4: (6 + 8+ 6 marks) a. Divide: *3-27 9-x2 x2+3x+9 x2 +9x+18 b. Solve: V3x + 2-27x=0 C. Solve: 3x7 - 24 x4=0

The expression can be expressed as a simplified power series in 2 on interval I as:
n=0 2^n*t^n [an+2 + 10an+1 + an]

To express the given expression as a simplified power series in 2 on interval I, we need to find the derivatives of y and substitute them into the expression.

First, we find the derivatives of y:

y' = an(nx^(n-1)) = nanx^(n-1)

y" = nan(n-1)x^(n-2)

Substituting y', y", and y into the given expression, we get:

(5 - 4x)(nan(n-1)x^(n-2)) + (2x)(nanx^(n-1)) + 3(anx^n)

= 5nan(n-1)x^n - 4nan(n-1)x^(n+1) + 2nanx^(n+1) + 3anx^n

Now we can express this as a power series in 2 by substituting x = 2t:

= 5nan(n-1)(2t)^n - 4nan(n-1)(2t)^(n+1) + 2nan(2t)^(n+1) + 3an(2t)^n

= 5nan(n-1)2^n*t^n - 8nan(n-1)2^(n+1)t^(n+1) + 2nan2^(n+1)t^(n+1) + 3an2^n*t^n

= 2^n*t^n [5nan(n-1) - 8nan(n-1)2t + 2nan(2t) + 3an]

= 2^n*t^n [an+2 + 10an+1 + an]

Therefore, the given expression can be expressed as a simplified power series in 2 on interval I as:

n=0 2^n*t^n [an+2 + 10an+1 + an]

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Answer:

x = 10.1, AXY = 71.7 degrees

Step-by-step explanation:

There are two ways to solve this problem. You could either do 7x+1+108.3=180 or 180-108.3, then take that number and set it equal to 7x+1.

I will be using the latter. (You can do this because angle YXB is a linear pair with angle AXY. This means they add up to 180. So to find angle AXY, you subtract 180 from 108.3)

[tex]180-108.3=71.7\\\\7x+1=71.7\\\\[/tex]

Subtract one from each side to move variables to the left and constants to the right.

[tex]7x+1-1=71.7-1\\\\7x=70.7[/tex]

Divide seven by both sides to isolate the variable.

[tex]\frac{7x}{7}=\frac{70.7}{7} \\\\x=10.1[/tex]

So now we know what x is. So to find AXY, you substitute it back into the equation.

[tex]7(10.1)+1=71.7\\\\70.1+1=71.7?\\\\71.1=71.7?[/tex]

Han's retirement account earns $247,163.25 in simple interest.

To find the simplest interest, we need to use the formula:

Simple Interest = Principal × Rate × Time

In this case, the Principal is $410,000 and the Rate is $15,785 per year. We don't know the time period, but we can solve for it using the formula:

Time = Simple Interest ÷ (Principal × Rate)

Plugging in the values, we get:

Time = $15,785 ÷ ($410,000 × 1) = 0.0385 years

Therefore, the simplest interest is:

Simple Interest = $410,000 × $15,785 × 0.0385 = $247,163.25

So Han's retirement account earns $247,163.25 in simple interest.

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The statement "Given the logistic function 3 x(t) = e-1.08 t +0.09 The time needed to reach x(t)= 98 is t=3." is :

(b) False

The logistic function is a mathematical function that is used to model growth processes that are limited by saturation. It is often used in the field of biology to model population growth, as well as in economics to model the growth of markets and the adoption of new technologies.

Given the logistic function x(t) = 3e^(-1.08t) + 0.09, you want to determine if x(t) = 98 when t = 3.

Step 1: Plug in t = 3 into the function
x(3) = 3e^(-1.08*3) + 0.09

Step 2: Calculate the result
x(3) ≈ 3e^(-3.24) + 0.09 ≈ 0.0705

Since x(3) ≈ 0.0705 and not 98, the statement "The time needed to reach x(t) = 98 is t = 3" is false. Therefore, the correct answer is:

b. False

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Answer:

Original Money Earned: $10,000

To increase this by 5 percent, we need to multiply $10,000 by 0.05 (5%).

$10,000 x 0.05 = $500

Allison makes $500 (5% of $10,000) per month, so you would add that to the sum of your answer after every previous month.

Now, let's add that.

Feb : $10,000 + 500 = $10,500

Mar : $10,500 + 500 = $11,000

Apr : $11,000 + 500 = $11,500

May : $11,500 + 500 = $12,000

Jun : $12,000 + 500 = $12,500

Jul : $12,500 + 500 = $13,000

Aug : $13,000 + 500 = $13,500

Sep : $13,500 + 500 = $14,000

Oct : $14,000 + 500 = $14,500

Nov : $14,500 + 500 = $15,000

Dec : $15,000 + 500 = $15,500

Allison can fill a series with a trend function in excel to calculate monthly income.

To calculate Allison's monthly income from February to December using Excel, you can use the fill series with a trend function.

1. Open a new Excel spreadsheet.
2. In cell A1, type "January" and in cell B1, type "$10,000" (without quotes) as Allison's January income.
3. In cell A2, type "February".
4. In cell B2, type the formula "=B1*1.05" (without quotes). This formula calculates the income for February by increasing January's income by 5 percent.
5. Click on cell B2 to select it, then move your cursor to the bottom right corner of the cell until the cursor changes into a small black cross.
6. Click and hold the left mouse button, then drag the cursor down to cell B12, which corresponds to December.
7. Release the left mouse button. Excel will fill the series with a trend, calculating the income for each month from February to December.

Hence, Excel is used to calculate Allison's monthly income from February to December, taking into account the expected 5 percent increase per month.

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the biology professor cannot conclude that the percentage of children who develop asthma in the new country is lower than the percentage observed in the US study.

The biology professor can use hypothesis testing to determine if the percentage of children who develop asthma in the new country is significantly different from the percentage observed in the US study.

Here are the steps she can take:

1. Define the null and alternative hypotheses:
- Null hypothesis (H0): The percentage of children who develop asthma in the new country is the same as the percentage observed in the US study (i.e., 22.1%).
- Alternative hypothesis (Ha): The percentage of children who develop asthma in the new country is lower than the percentage observed in the US study.

2. Determine the test statistic to use:
- The appropriate test statistic for this scenario is the one-sample proportion z-test.

3. Set the significance level (alpha):
- Let's assume a significance level of 0.05.

4. Calculate the test statistic:
- The sample proportion of children who developed asthma in the new country is p = 70/336 = 0.2083.
- The standard error of the sample proportion is SE = sqrt[(p*(1-p))/n] = sqrt[(0.2083*(1-0.2083))/336] = 0.027.
- The test statistic is z = (p - P) / SE, where P is the proportion observed in the US study. So, z = (0.2083 - 0.221) / 0.027 = -0.463.

5. Determine the p-value and make a decision:
- The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. Using a standard normal distribution table or calculator, we find that the p-value is 0.3212.
- Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the percentage of children who develop asthma in the new country is significantly different from the percentage observed in the US study.

Therefore, the biology professor cannot conclude that the percentage of children who develop asthma in the new country is lower than the percentage observed in the US study.

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Answer:

[tex].12 \times 360 \: degrees = 43.2 \: degrees[/tex]

The manager of a laptop computer dealership is considering a new bonus plan in order to increase sales. Currently, the mean sales rate per salesperson is five laptops per week.H0 (null hypothesis) represents the current situation, which is the mean sales rate per salesperson being 5 laptops per week. Ha (alternative hypothesis) represents the expected change, which is an increase in the mean sales rate due to the bonus plan.

The correct set of hypotheses for testing the effect of the bonus plan in this scenario is option c: H0: μ ≤ 5 Ha: μ > 5.
This is because the manager wants to increase sales, which means they are hoping for a higher mean sales rate per salesperson. Therefore, the null hypothesis (H0) is that the mean sales rate is less than or equal to 5 (the current rate), while the alternative hypothesis (Ha) is that the mean sales rate is greater than 5.

Option a (H0: μ < 5 Ha: μ ≥ 5) and option d (H0: μ > 5 Ha: μ < 5) both assume that the manager wants to maintain the current sales rate or decrease it, which is not the case. Option b (H0: μ > 5 Ha: μ < 5) assumes that the manager wants to decrease the sales rate, which is also not the case.

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The equations are;

H = 24 + 9x

H = 40 + 5x

In a word problem, there are usually one or more unknown quantities that you need to find. Identify these unknowns and assign them a variable.

We have to know that Tree A was 24 inches tall and Tree B was 40 inches tall. Each year thereafter, Tree A grew by 9 inches per year and Tree B grew by 5 inches per year.

Then for tree A;

H = 24 + 9x

For tree B

H = 40 + 5x

Where x is the number of years that the trees stay.

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The standardized residuals are -0.11, 0.75, -0.38, 2.08, and 2.16, respectively.

(a) The estimated regression equation can be found by first calculating the sample means and sample standard deviations for both x and y, as well as the sample correlation coefficient, and then using these values to calculate the slope and intercept of the regression line:

x = (6 + 11 + 15 + 18 + 20)/5 = 14

y = (7 + 7 + 13 + 21 + 30)/5 = 15.6

sx = sqrt(((6-14)^2 + (11-14)^2 + (15-14)^2 + (18-14)^2 + (20-14)^2)/4) = 4.49

sy = sqrt(((7-15.6)^2 + (7-15.6)^2 + (13-15.6)^2 + (21-15.6)^2 + (30-15.6)^2)/4) = 8.25

r = ((6-14)(7-15.6) + (11-14)(7-15.6) + (15-14)(13-15.6) + (18-14)(21-15.6) + (20-14)(30-15.6))/4sxsy

= 0.962

b = r(sy/sx) = 1.71

a = y - b(x) = -9.23

Therefore, the estimated regression equation is y = -9.23 + 1.71x.

(b) The residuals can be found by subtracting the predicted values of y (based on the regression line) from the actual values of y:

xi

yi

y

Residuals

6

7

0.09

-0.09

11

7

6.38

0.62

15

13

13.31

-0.31

18

21

19.29

1.71

20

30

23.57

6.43

(c) The standardized residuals can be found by dividing each residual by its estimated standard deviation:

xi

yi

ŷ

Residuals

Standardized Residuals

6

7

0.09

-0.09

-0.11

11

7

6.38

0.62

0.75

15

13

13.31

-0.31

-0.38

18

21

19.29

1.71

2.08

20

30

23.57

6.43

2.16

The estimated standard deviation of the residuals can be found by calculating the root mean squared error (RMSE):

RMSE = sqrt(((-0.09)^2 + (0.62)^2 + (-0.31)^2 + (1.71)^2 + (6.43)^2)/4) = 3.04

Therefore, the standardized residuals are -0.11, 0.75, -0.38, 2.08, and 2.16, respectively.

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The number is -4. We can solve for it using algebraic equations and multiplication.

The problem states that when we multiply our number by four and add twenty, we get 4. We can represent this relationship using an algebraic equation with the variable x representing our unknown number:

4x + 20 = 4

To solve for x, we need to isolate it on one side of the equation. We start by subtracting 20 from both sides of the equation:

4x = -16

Now, we divide both sides by 4:

x = -4

Therefore, our number is -4. We can check this answer by plugging it back into the original equation:

Simplifying the left side of the equation, we get:

-16 + 20 = 4

4 = 4

This confirms that our answer, x = -4, is correct.

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Based on the information, we can see from the line plots that Bus 14 tends to have longer travel times than Bus 18 for most of the data points, except for a few outliers.

In terms of travel time, Bus 14 and Bus 18 each have a median of 16 minutes. As such, it cannot be inferred from this information alone which mode of transportation tends to arrive more rapidly.

Nevertheless, the line plots reveal that Bus 14's journey takes slightly longer than Bus 18's for most of the data points, except for a few outliers.

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The surface area of the composite figure is 1474 square feet.

In the composite figure, there are two shapes rectangular prism and triangular prism.

Surface area of rectangular prism = 2(lb+bh+hl)

= 2(19×9+9×11+11×19)

= 958 square feet

Surface area of triangular prism = (Perimeter of the base × Length of the prism) + (2 × Base Area)

= (S1 +S2 + S3)L + bh

= (13+13+10)×11+10×12

= 516 square feet

Total surface area = 958+516

= 1474 square feet

Therefore, the surface area of the composite figure is 1474 square feet.

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The correct test statistic for Amanda's significance test would be option B:
z = (0.49 - 0.60) / sqrt(0.49(0.51)/300)

This is because option B includes the sample proportion and the sample size, which are necessary for calculating the test statistic for a significance test involving proportions. The formula for the test statistic for a two-tailed test of a population proportion is:

z = (p - P) / sqrt(P(1 - P) / n)

where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.

In this case, we are not given the hypothesized population proportion, so we use the sample proportion as an estimate. The formula becomes:

z = (p - P) / sqrt(P(1 - P) / n) = (p - 0.5) / sqrt(0.5(0.5) / n) = (0.49 - 0.5) / sqrt(0.5(0.5) / 300)

Simplifying this expression gives us the test statistic in option B.

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To find out how many students prefer Kiwi when there are 357 total students, we can use the equivalent ratios of 5:12 or 12:5.

The ratio of students who prefer pineapple to students who prefer kiwi is given as 12 to 5, which means that for every 12 students who prefer pineapple, 5 students prefer kiwi. We can represent this ratio as 12:5.

To find out how many students prefer kiwi, we need to determine the proportion of the total number of students that prefer kiwi. Since the total number of students is 357, we can set up a proportion with the ratio of students who prefer Kiwi to the total number of students. Using the equivalent ratio of 5:12, we can set up the proportion as follows:

5/12 = x/357

Here, x represents the number of students who prefer Kiwi. To solve for x, we can cross-multiply and simplify the proportion as follows:

5 * 357 = 12 * x
1785 = 12x
x = 1785/12
x = 148.75

Since we cannot have a fractional number of students, we need to round our answer to the nearest whole number. Therefore, we can conclude that approximately 149 students prefer Kiwi out of a total of 357 students.

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Answer:

V = 58.7 m^3

Step-by-step explanation:

The formula is V = 1/3 *b*h

where b is the base area

h is the height

From the diagram,

h= 11m

Now we need to find the area of the base.

The base is made of an equilateral triangle, so we know that one of the sides is 8m.

The area of the base is found: 1/2 * a * l

Where a = 8m

l = a/2 = 8m/2 = 4m

So the area of the base is:

b = 1/2 * 8 m* 4m

b = 16m^2

Now plug this into the volume formula:

V = 1/3 b * h

V = 1/3 * 16m^2 * 11m

V = 58.7 m^3

The probability of a point chosen uniformly at random in the region [tex]-1\leq x\leq1[/tex] and [tex]-1\leq y\leq1[/tex]  to be classified as '+' is [tex](\frac{4 - A_{minus}}{4})[/tex].

To draw the Voronoi tile for the given data set and find the probability of a point being classified as '+', follow these steps:

1. Plot the data points: Plot the points (-1,-1), (-1,1), (1,-1), (1,1), and (0,0) on a graph. Label (0,0) as '-' and the other points as '+'.

2. Construct the Voronoi diagram:

For each pair of neighboring '+' points, draw a line that is equidistant from both points and bisects the line connecting them. These lines will divide the space into regions called Voronoi tiles, where each tile contains one data point, and any point within that tile is closer to the data point it contains than to any other data point.

3. Identify the tile containing the '-' point:

In this case, the Voronoi tile surrounding (0,0) will be the region that is closer to the '-' point than to any '+' point.

4. Calculate the area of the Voronoi tile containing the '-' point:

Since we are considering the region [tex]-1\leq x\leq1[/tex] and [tex]-1\leq y\leq1[/tex], find the area of the intersection of this region with the Voronoi tile containing the '-' point.

5. Calculate the total area of the considered region: The total area of the considered region is

(-1 to 1)(-1 to 1) = 2(2) = 4 square units.

6. Determine the probability of a point being classified as '+':

The probability of a point chosen uniformly at random in the considered region being classified as '+' is equal to the ratio of the area of the region not covered by the '-' Voronoi tile to the total area of the considered region.

Let's say the area of the '-' Voronoi tile is [tex]A_{minus}[/tex]. Then, the probability of a point being classified as '+' is [tex](\frac{4 - A_{minus}}{4})[/tex].

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Answer: ''pencil veinya''    (Pennsylvania)

Step-by-step explanation:

The value of angle O is 11 degrees

It is important to note that the properties of a triangle are;

From the information given, we have the angles;

m<N = 5x - 8

m<O = x - 5

m<P = 6x + 1

Also, the sum of the angles in a triangle is 180 degrees

Now, substitute the angles

m<O + m<P + m<O = 180

5x - 8 + x - 5 + 6x + 1 = 180

collect the like terms

5x + x + 6x = 180 + 12

add the terms

12x = 192

x = 16

For the angle, m<O = x - 5 = 16 -5 = 11 degrees

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The value of the expectation E[aX1 + bX2 + cX3] is aE[X1] + bE[X2] + cE[X3]

The value of the variance Var[aX1 + bX2 + cX3] is a² Var[X1] + b² Var[X2] + c² Var[X3]

We have,

Using the linearity of expectation and the fact that the variables are independent, we have:

E[aX1 + bX2 + cX3]

= aE[X1] + bE[X2] + cE[X3]

And for the variance, using the fact that the variables are independent and using the property Var[aX] = a^2 Var[X], we have:

Var[aX1 + bX2 + cX3]

= Var[aX1] + Var[bX2] + Var[cX3]

= a^2 Var[X1] + b^2 Var[X2] + c^2 Var[X3]

Note that we have used the fact that the covariance between any two distinct X_i, X_j is zero since they are independent,

i.e., Cov[X_i, X_j] = E[X_iX_j] - E[X_i]E[X_j] = E[X_i]E[X_j] - E[X_i]E[X_j] = 0.

Thus,

The value of the expectation E[aX1 + bX2 + cX3] is aE[X1] + bE[X2] + cE[X3]

The value of the variance Var[aX1 + bX2 + cX3] is a² Var[X1] + b² Var[X2] + c² Var[X3]

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The graph of the function g(x) = −|x + 3| − 2 is added as an attachment

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

g(x) = −|x + 3| − 2

The above expression is an absolute value function that hs the following properties

Next, we plot the graph

See attachment for the graph of the function

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For a matrix, [tex]A = \begin{bmatrix} 3 & 2& 4 \\ 1& -2& 3\\ 2 &3 &2 \end{bmatrix}\\[/tex]

a) The value of det(M₂₁), det(M₂₂), and det(M₂₃) are -8, -2 and 5 respectively.

b) The values of cofactors of matrix A, A₂₁, A₂₂ and A₂₃ are 8, -2 and -5 respectively.

c) The computed value of determinant, det(A) is equals to -3.

Matrix is a set of elements arranged in rows and columns in order to form a rectangular array. We have a matrix A, defined as [tex]A = \begin{bmatrix} 3 & 2& 4 \\ 1& -2& 3\\ 2 &3 &2 \end{bmatrix}\\[/tex]

We have to determine the following values :

a) The minor of matrix is exist for each element of matrix and is equal to the part of the matrix remained after removing the row and the column containing. First we determine the value Minors and then their determinant. [tex]M_{21} = \begin{bmatrix} 2& 4 \\ 3& 2\\ \end{bmatrix}\\[/tex]

det(M₂₁ ) = | M₂₁ | = 2× 2 - 4× 3 = - 8

[tex]M_{22} = \begin{bmatrix} 3& 4 \\ 2& 2\\ \end{bmatrix}\\[/tex]

det(M₂₂) = | M₂₂| = 2× 3 - 4× 2 = - 2

[tex]M_{23} = \begin{bmatrix} 3& 2 \\ 2& 3\\ \end{bmatrix}\\[/tex]

det(M₂₃ ) = | M₂₃ | = 3×3 - 2×2= 5

b) The value of cofactors of matrix A are A₂₁ = (-1)²⁺¹ M₂₁

= -(-8) = 8

A₂₂ = (-1)²⁺² M₂₂ = -2

A₂₃ = (-)²⁺³ M₂₃ = -5

c) Now, we determine the value of determinant of matrix A from part (b).

det(A) = a₂₁ A₂₁ + a₂₂ A₂₂ + a₂₃ A₂₃

= 1× 8 - 2 (-2) + 3( -5)

= -3

Hence, required value is -3.

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Answer:

When data has a narrow range and we want to see the shape of the data, the best way to display the data is through a line plot. A line plot is a graphical display of data that uses dots placed above a number line to show the frequency of values in a set of data. It is useful for showing the distribution of a small set of data, especially when the data has a narrow range of values. The line plot allows us to see how many times each value occurs and to identify the mode(s) of the data.

When data has a narrow range and we want to see its shape, a stem-and-leaf plot is the best way to display it. In a stem-and-leaf plot, the data is divided into two parts: the stem and the leaf. The stem is the leftmost digit(s) of each data point, and the leaf is the rightmost digit. This plot allows us to quickly see the distribution of the data and identify any outliers or patterns.

The solution to the given quadratic equation is x= -3 -2 -1 0 1 2 3 y= 2 3 2 -1 -6 -13 -22.

This table of values contains a quadratic function that changes direction at the value of x = 0. This is different from the other tables of values which all have the quadratic function changing direction at the value of x = -2. The y values in this table of values can be described as an upside-down parabola. At x = 0, the y value is -1 and it decreases as x increases to positive values, and it increases as x decreases to negative values.

Therefore correct answer is C. x -3 -2 -1 0 1 2 3 y 2 3 2 -1 -6 -13 -22.

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With 99% confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval.

Part 1:

To construct a 99% confidence interval for the population proportion, we can use the following formula:

CI = P ± z*√(P(1-P)/n)

where P is the sample proportion, z is the z-score corresponding to the desired level of confidence (in this case, 99%), and n is the sample size.

In this case, P = 1446/2728 = 0.5298, z = 2.576 (from a standard normal distribution table), and n = 2728. Plugging these values into the formula, we get:

CI = 0.5298 ± 2.576√(0.5298(1-0.5298)/2728)

CI = (0.5085, 0.5511)

So the 99% confidence interval for the population proportion is (0.5085, 0.5511).

Part 2:

The correct interpretation is A. With 99% confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval.

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The percentage difference in the percent errors of the estimates is approximately 35.6%.

Let's start with the drywall estimate. The estimated value is 19 sheets, while the actual value is 16 sheets. Using the formula, we get:

Percent Error = (|19 - 16| / 16) x 100%

Percent Error = (3 / 16) x 100%

Percent Error = 18.75%

Therefore, the percent error in the drywall estimate is 18.75%.

Now, let's calculate the percent error in the tile estimate. The estimated value is 85 square feet, while the actual value is 67 square feet. Using the formula, we get:

Percent Error = (|85 - 67| / 67) x 100%

Percent Error = (18 / 67) x 100%

Percent Error = 26.87%

Therefore, the percent error in the tile estimate is 26.87%.

To find the difference in the percent errors, we need to subtract the percent error in the drywall estimate from the percent error in the tile estimate and take the absolute value. We then divide the result by the average of the percent errors and multiply by 100 to get the percentage difference. The formula is as follows:

Percentage Difference = |(Percent Error Tile - Percent Error Drywall) / ((Percent Error Tile + Percent Error Drywall) / 2)| x 100%

Plugging in the values, we get:

Percentage Difference = |(26.87% - 18.75%) / ((26.87% + 18.75%) / 2)| x 100%

Percentage Difference = |8.12% / 22.81%| x 100%

Percentage Difference = 35.6%

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The printer that has a better rate of cost per page would be = printer B.

For printer A:

The number of pages printed = 100

The cost for the printed pages = $26.99

Cost per page = 26.99/100 = $0.27/page

For printer B:

The number of pages printed = 275

The cost for the printed pages = $67.99

Cost per page =67.99/275

= 0.25

Therefore the printer that has a better rate of cost per page would be = printer B

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