As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right?

The integral is [tex]\int\limits^2_3 {\sqrt{(1 + (12x^3 + 2)^2)} } dx[/tex] represents the arc length of the curve y = 3x^4+2x from (-2,-52) to (3,249).

To find the arc length of the curve y = 3x^4+2x from (-2,-52) to (3,249), we need to set up the integral of the arc length formula:

[tex]\int\limits^a_b {\sqrt{√(1 + (dy/dx)^2) } } \, dx[/tex]

where a and b are the x-coordinates of the starting and ending points, respectively.

First, we need to find dy/dx:

[tex]dy/dx = 12x^3 + 2[/tex]

Next, we need to plug in the values of the starting and ending points to get a and b:

a = -2, b = 3

Now, we can substitute the values of dy/dx, a, and b into the arc length formula:

[tex]L=\int\limits^2_3 {\sqrt{(1 + (12x^3 + 2)^2)} } dx[/tex]

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You can sketch the curve with x-intercepts at x = -5, 0, and 5, y-intercept at f(0) = 0, and considering the end behavior and symmetry.

To sketch the curve of the function f(x) = x(x² - 25), follow these steps:

1. Identify the type of function: This is a cubic function because the highest power of x is 3 (x × x²).
2. Determine the x-intercepts: To find the x-intercepts, set f(x) to 0 and solve for x.
  0 = x(x² - 25) => x = 0, x² = 25 => x = -5, x = 5
3. Determine the y-intercept: To find the y-intercept, set x to 0 and solve for f(x).
  f(0) = 0(0² - 25) => f(0) = 0
4. Identify any symmetry: As the function is odd (cubic), there is rotational symmetry around the origin.
5. Check for end behavior: As this is a cubic function with a positive leading coefficient, the function will approach negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity.

Now, you can sketch the curve with x-intercepts at x = -5, 0, and 5, y-intercept at f(0) = 0, and considering the end behavior and symmetry.

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The area of given quadrilateral is 88 sq. units, as the quadrilateral with points (2,10) ; (-4,10) ; (-6,-1) and (4,-1) when plotted on graph represents the trapezium.

What is a trapezium?

The trapezium is a convex, two-dimensional quadrilateral with one pair of opposite sides that are perfectly parallel to one another. A two-dimensional object called a trapezium has one pair of parallel opposed sides and four angles, four vertices, and four sides. The bases and the non-parallel sides of a trapezium are referred to as sides of a trapezium, respectively.

Area of trapeziod=[tex]\frac{(a+b)h}{2}[/tex]   {where a & b are parallel sides & h is the perpendicular height between them}

Dimensions of given quadrilateral: parallel sides= 6 units & 10 units

height=11 units

Area of quadrilateral= area of trapezium

                                 =[tex]\frac{(a+b)h}{2}[/tex]  

                                 =[tex]\frac{(6+10)11}{2}[/tex]

                                 =[tex]\frac{11(16)}{2}[/tex]

                                 =11 x 8

                                  =88 sq. units

Area of quadrilateral=88 sq. units.

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Here since the ratios are not all equal, we can conclude that x and y are not in a proportional relationship.

What is meant by ratios?

Ratios are a mathematical tool used to compare the sizes of two or more quantities, expressed as a ratio of their values. Ratios can be simplified, converted to fractions or decimals, and used to solve problems in various fields.

What is meant by proportional?

Proportional refers to the constant relationship between two or more quantities where their ratio remains the same. Scaling one quantity by a certain factor requires scaling the other quantity by the same factor to maintain the proportion.

According to the given information

To determine if x and y are in a proportional relationship, we need to check if the ratio of y to x is constant.

Let's calculate the ratio of y to x for each pair of corresponding values:

y/x for (1, 2.5) = 2.5/1 = 2.5

y/x for (2, 5) = 5/2 = 2.5

y/x for (3, 7.5) = 7.5/3 = 2.5

y/x for (6, 12) = 12/6 = 2

Since the ratios are not all equal, we can conclude that x and y are not in a proportional relationship.

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Answer:

B. -3/2

Step-by-step explanation:

Slope is measured by rise over run

It moves 3 units up and 2 units to the left

The distance is the square root of the sum of the squares of the differences in height and age, which gives SQRT(121 +36) = SQRT(157) = 12.53. So, correct option is A.

The Euclidean distance between two points is calculated using the Pythagorean theorem, which gives the distance between the two points as the square root of the sum of the squares of the differences in their coordinates.

In this case, the two people are characterized by their height and age, so we need to calculate the distance between the points (191, 24) and (180, 18).

Option (A) correctly calculates the Euclidean distance between the two points using the Pythagorean theorem. The distance is the square root of the sum of the squares of the differences in height and age, which gives SQRT(121 +36) = SQRT(157) = 12.53.

Option (B) calculates the Manhattan distance between the two points, which is the sum of the absolute differences in height and age. This is not the Euclidean distance.

Option (C) incorrectly swaps the coordinates of the two points and calculates the Euclidean distance between them. This is not the distance between the two people.

Therefore, the correct answer is (A), which calculates the Euclidean distance between the two points using the Pythagorean theorem.

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If fixed costs are $10, the minimum average total cost occurs at output of value 4.

To find the minimum average total cost, we need to calculate the average cost for each level of output. The average cost is the sum of total costs (fixed cost plus variable cost) divided by the output level.

The variable cost can be calculated by finding the difference between the total cost of the previous level and the total cost of the current level.

Output level | Variable cost | Total cost | Average cost

0                    | 0                    | 10              | -

1                     | 15                   | 25             | 25

2                    | 10                   | 35             | 17.5

3                    | 5                     | 40            | 13.3

4                    | 8                     | 48            | 12

5                    | 12                   | 62            | 12.4

6                    | 20                  | 90            | 15

As we can see from the table, the minimum average total cost occurs at an output level of 4, which has an average cost of $12 per cake. Therefore, the answer is 4.

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Complete question is:

Costs of birthday Cakes                    Variable Cost,

0                                                                         0

1                                                                           15

2                                                                           25

3                                                                           30

4                                                                            38

5                                                                             50

6                                                                             70

assume that fixed costs are $10. the minimum average total cost occurs at output of:

K is greater than 1.96 in both cases. This means that the interval of Values from -1.96 to 1.96 covers 95% of the area under the normal distribution curve, and the probability of a value falling outside this interval is 0.05.

To solve this question, we need to use the standard normal distribution table or calculator.

First, we need to find the critical values for the given confidence level and sample size. For a 95% confidence level and a sample size of 24, the critical values are ±2.069.

To find k such that p(−2.069 < Z < k) = 0.47, we need to look up the area of 0.47 in the standard normal distribution table. The closest area we can find is 0.475, which corresponds to a z-score of 1.96. Therefore, we have:

−2.069 < Z < k
−2.069 < Z < 1.96

To solve for k, we need to isolate it in the inequality. Adding 2.069 to both sides, we get:

0 < Z + 2.069 < k + 2.069
0 < 1.96 + 2.069 < k + 2.069

Simplifying:

4.029 < k + 2.069

Subtracting 2.069 from both sides:

1.96 < k

Therefore, k is greater than 1.96.

To find k such that p(−k < Z < k) = 0.95, we need to find the z-score that corresponds to the area of 0.475 on either side of the mean. Using the standard normal distribution table, we find that the z-score is 1.96. Therefore, we have:

−k < Z < k

To solve for k, we need to isolate it in the inequality. Adding k to both sides, we get:

0 < Z + k < 2k
0 < 1.96 + k < 2k

Simplifying:

1.96 < k

Therefore, k is greater than 1.96.

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Step-by-step explanation:

Mean is the sum of all values divided by the total number of values

Median is the value in the middle of a data set

(the median is the middle value when the data set is an odd number, or the arithmetic mean of the two medians when the data set is an even number)

Mode is the most frequent value

Range is the difference between the highest and the lowest values in a data set

.

1. Mean:

[tex] \frac{3 + 0 + 0 + 2 + 0 + 3 + 0 + 2 + 2 + 2 + 3 + 3}{12} = 1 \frac{2}{3} [/tex]

Median: The data set is an even number (12)

The middle values are 3 and 0

[tex] \frac{3 + 0}{2} = 1.5[/tex]

Modes: 2 and 3

Range: The highest value is 3 and the lowest is 0

[tex]3 - 0 = 3[/tex]

.

2. Mean:

[tex] \frac{40 + 61 +95 + 79 + 9 +50 + 80 + 63 + 109 + 42 }{10} = 62.8[/tex]

Median: The data set is an even number (10)

The middle values are 5 and 6

[tex] \frac{5 + 6}{2} = \frac{11}{2} = 5.5[/tex]

Mode: There is no mode

Range: The highest value is 109 and the lowest is 9

[tex]109 - 9 = 100[/tex]

.

3. Mean:

[tex] \frac{90 + 50 + 70 + 80 + 70 +60 + 20 + 30 + 80 + 90 + 20}{11} = 60[/tex]

Median:The data set is an odd number (11)

The middle value is 6

Modes: 20, 70, 80, 90

Range: The highest value is 90 and the lowest is 20

[tex]90 - 20 = 70[/tex]

.

4. Mean:

[tex] \frac{98 + 100 +65 + 78 +98 + 35 + 100 + 45 + 50 }{9} = 74 \frac{1}{3} [/tex]

Median: The data set is an odd number (9)

The middle value is 98

Modes: 100 and 98

Range: The highest value is 100 and the lowest is 35

[tex]100 - 35 = 65[/tex]

.

5. Mean:

[tex] \frac{8 + 2 + 9 +4 + 2 +7 + 8 + 0 + 4 + 1}{10} = 4.5[/tex]

Median: The data set is an even number (10)

The middle values are 2 and 7

[tex] \frac{2 + 7}{2} = 4.5[/tex]

Modes: 2, 4, 8

Range: The highest value is 9 and the lowest is 0

[tex]9 - 0 = 9[/tex]

.

6. Mean: 22 1/15

Median: The data set is an even number (30)

The middle values are 21 and 23

The median is (21 + 23) / 2 = 22

Mode: 24 and 32

Range: The highest value is 40 and the lowest is 6

40 - 6 = 34

On solving the mentioned equation, the correct answer for value of x will be d. no solution.

We will begin with converting the base to same numbers. As per the fact, 49 is the square of 7 and 323 is the cube of 7. So, the equation will be -

[tex] {7²}^{3x} = {7³}^{(2x + 1)} [/tex]

Now, performing the multiplication of both exponents on both sides of the equation

[tex] {7}^{6x} = {7}^{(6x + 3)} [/tex]

As the bases are equal, hence will be the exponents.

6x = 6x + 3

6x is common on both sides and hence x will not exist. Thus, no solution is possible.

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The correct question is -

Solve: [tex] {49}^{3x} = {343}^{(2x + 1)} [/tex]. a. x = –3 b. x = 1 c. x = 3 d. no solution

Answer:

It's D, No solution.

Step-by-step explanation:

I did the assignment.

The proportion of students in our sample earned B’s on the final exam is 0.4572. The 90% confidence interval for B scorer is (0.4083, 0.5061).

a) To obtain the number of B's, we need to count the number of scores that fall between 64 and 71 inclusive. Using the "Interactive Tools" option in StatCrunch, we can select "Scores" as the variable and slide the range selector to select scores between 64 and 71. The number of rows selected is 123, which represents the number of students who earned a Bin the sample. The proportion of students who earned a B is 123/269 = 0.4572 (rounded to four decimal places).

c) Using the StatCrunch function "Proportions Stats" -> "One Sample" -> "With Summary", we can enter the sample size (269) and the number of students who earned a B (123) to obtain a 90% confidence interval. The resulting interval is (0.4086, 0.5059), which is very close to our earlier calculation.

d) We are 90% confident that the true proportion of students who earned a B on the final exam falls between 0.4086 and 0.5059.

e) No, this confidence interval does not capture the true population proportion p = 0.21 given in Problem 4 of Data Analysis 2, as the interval (0.4086, 0.5059) does not contain 0.21.

f) Using the Confidence Interval applet in StatCrunch, we can simulate constructing 1000 90% confidence intervals for the population proportion with p = 0.21 and n = 269. The resulting plot shows that the proportion of intervals that contain the true population proportion is about 0.88.

g) The "Prop. contained" value from part (f) is 0.88, which is the proportion of intervals that contain the true population proportion. This value is equivalent to the confidence level associated with the simulation, which is 90%. This suggests that our confidence interval from part (b) is a reasonable estimate of the population proportion of the sample.

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Step-by-step explanation:

circumference= 2× radius

circumference of ceiling fan

= 2×2.8ft

= 5.6ft

circumference of water bottle cap

= 2×13mm

= 26mm

radius of bowl

= 56.5cm÷2

= 28.25cm

radius of drum

= 75.4in÷2

= 37.7in

When a population is growing logistically the change in population size (dn/dt) is greatest  is given by  option c. when k/2 is equal to n.

Population size = n

Time = t

Change in population size over time = dn/dt

Option a,

This occurs because when the population is small .

There are fewer individuals to reproduce and contribute to the population growth.

Option b,

As the population approaches the carrying capacity  the growth rate slows down.

Because of environmental limitations on resources, space, and other factors.

At the carrying capacity, the population growth rate is zero, and the population remains stable.

The fastest population growth rate occurs when the population size is at an intermediate level.

Around half of the carrying capacity.

When the population size n is approximately half of the carrying capacity k of the environment.

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Answer:

right 3 units

Step-by-step explanation:

for a translation following f(x-a), the graph is shifted right by 'a' units; so 'a' is added to the x coordinates. so for f(x-3), 3 is added to the x coordinates, meaning that it shifts to the right by 3 units.

for future reference for similar questions:

if it was f(x+a), then it would shift to the left by 'a' units

if it was f(x)+a, then it would shift up by 'a' units

if it was f(x)-a, then it would shift down by 'a' units

if it was -f(x), then it would be reflected on the x axis

if it was f(-x), then it would be reflected on the y axis

hope this helps!

The given expression can be rewritten as, [tex]$\sum_{n=0}^{\infty} (4n-7)c_n x^n$[/tex], where the coefficient [tex]$c_k$[/tex] is given by, [tex]$c_k = \begin{cases}\frac{4k-7}{k} c_k & \text{if } k > 0 \c_0 & \text{if } k = 0\end{cases}$[/tex].

In Example 3 of Section 6.1, we showed how to rewrite the expression

[tex]$\sum_{n=1}^{\infty} \frac{x^n}{n(n+1)}$[/tex]

using a single power series with general term involving [tex]$x^k$[/tex]. We'll use the same approach here to rewrite the expression

[tex]$\sum_{n=1}^{\infty} 4nc_n x^{n-1} - \sum_{n=0}^{\infty} 7c_n x^n$[/tex]

as a single power series with general term involving [tex]$x^k$[/tex].

First, we'll rewrite the first sum with a shifted index so that the exponents of x start at 0:

[tex]$\sum_{n=1}^{\infty} 4nc_n x^{n-1} = \sum_{k=0}^{\infty} 4c_{k+1} (k+1)x^k$[/tex]

Next, we'll use the formula for the derivative of a power series to rewrite this as:

[tex]$\frac{d}{dx} \sum_{n=1}^{\infty} 4nc_n x^{n-1} = \sum_{k=0}^{\infty} 4c_{k+1} (k+1)x^k$[/tex]

Note that the sum on the left-hand side is the derivative of the power series [tex]$\sum_{n=0}^{\infty} 4nc_n x^n$[/tex], which we can write as:

[tex]$\frac{d}{dx} \sum_{n=0}^{\infty} 4nc_n x^n = \sum_{k=0}^{\infty} 4c_{k+1} (k+1)x^k$[/tex]

Combining this with the expression for the second sum, we get:

[tex]$\frac{d}{dx} \sum_{n=1}^{\infty} 4nc_n x^{n-1} - \sum_{n=0}^{\infty} 7c_n x^n = \frac{d}{dx} \sum_{n=0}^{\infty} 4nc_n x^n - \sum_{n=0}^{\infty} 7c_n x^n$[/tex]

Simplifying, we get:

[tex]$4\sum_{n=1}^{\infty} nc_n x^{n-1} - \sum_{n=0}^{\infty} 7c_n x^n = \sum_{n=0}^{\infty} (4n-7)c_n x^n$[/tex]

Therefore, the given expression can be rewritten as:

[tex]$\sum_{n=0}^{\infty} (4n-7)c_n x^n$[/tex]

which is a single power series with general term involving [tex]$x^k$[/tex], where the coefficient [tex]$c_k$[/tex] is given by:

[tex]$c_k = \begin{cases}\frac{4k-7}{k} c_k & \text{if } k > 0 \c_0 & \text{if } k = 0\end{cases}$[/tex]

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Answer:   Correct option is A)

The force of friction opposes the relative motion between the heavy box and rough surface.

Step-by-step explanation:

Answer:

4000 Newtons.

Step-by-step explanation:

1000 + 3000

Answer:

D-7.5

go to the $65 mark on the left side of the graph, then go straight till you hit the line. when you hit the line go straight down, which brings you to 7.5

you can rent a car for 7.5 hours for $65

make sense?

Answer:

y=6x+1

Step-by-step explanation:

Point slope equation is y-y¹=m(x-x¹)

Substitute your numbers in as y-19=6(x-3)

Distribute M to get y-19=6x-18

Add 19 to both sides to get y=6x+1

(a) About 38.21% of students at UPW University have a GPA of at least 3.

(b) Approximately 58.57% of students' GPAs are between 2 and 3.3.

a) To find the proportion of students with a GPA of at least 3, we use the z-score formula: z = (X - μ) / σ. Here, X = 3, μ = 2.88, and σ = 0.6. So, z = (3 - 2.88) / 0.6 = 0.2.

Using a z-table, we find that the proportion of students with a z-score below 0.2 is 0.5793. Since we want the proportion of students with a GPA above 3, we subtract this value from 1: 1 - 0.5793 = 0.4207, or about 38.21%.

b) For the proportion of students with a GPA between 2 and 3.3, we first calculate the z-scores for both values. For X = 2, z = (2 - 2.88) / 0.6 = -1.4667. For X = 3.3, z = (3.3 - 2.88) / 0.6 = 0.7.

Using the z-table, we find the proportions for z = -1.4667 and z = 0.7 as 0.0714 and 0.7580, respectively. Subtracting the lower proportion from the higher one gives 0.7580 - 0.0714 = 0.6866, or about 58.57%.

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Answer:

a=12

Step-by-step explanation:

(2a-1)+(6a-5)=90

First, we can simplify the left side by combining the two like terms (2a and 6a) and the two constant terms (-1 and -5):

8a - 6 = 90

Next, we can add 6 to both sides to isolate the variable term on one side of the equation:

8a = 96

Finally, we can divide both sides by 8 to solve for a:

a = 12

Therefore, the solution is a = 12.

*IG: whis.sama_ent*

Answer:

a = 12

Step-by-step explanation:

Angle (2a-1) and angle (6a-5) are complementary angles.

Complementary angles sum 90°

Then;

(2a-1) + (6a-5) = 90

2a + 6a -1 - 5 = 90

8a - 6 = 90

8a = 90 + 6

8a = 96

a = 96/8

a = 12

Check:

(2*12 - 1) + (6*12-5) = 90

24 - 1 + 72 - 5 = 90

Answer:

12 m²

Step-by-step explanation:

Let the perimeter of the rectangle be known as P, width of the rectangle be known as W, and the length of the rectangle be known as L.

Given Information:

We know that the perimeter is the sum of the boundaries of the rectangle. Therefore, the perimeter is equivalent to twice the sum of L and W.

Substitute the given variables into the perimeter equation;

Then, solve for the width of the rectangle;

Finally, solve for the area of the rectangle;

Therefore, the area of the rectangle is 12 m² (A)

we cannot determine if it converges or diverges analytically.

I assume you are asking about the integral of [tex]e^(xlnx)[/tex] from 3 to 13. To determine whether the integral is convergent or divergent, we can follow these steps:

1. Write the integral: ∫(3 to 13)[tex]e^(xlnx) dx[/tex]
2. Apply integration by parts: Let u = ln(x) and dv =[tex]e^(xlnx) dx[/tex]
3. Differentiate u and integrate dv: du = (1/x) dx, v = ∫ [tex]e^(xlnx) dx[/tex]
4. Apply integration by parts again to find v: Let u1 = x and dv1 = [tex]e^(xlnx) dx[/tex]
5. Differentiate u1 and integrate dv1: du1 = dx, v1 = ∫ [tex]e^(xlnx) dx[/tex]
6. Now, we have the following equation: v = xv1 - ∫ v1 du1

Unfortunately, the integral of[tex]e^(xlnx) dx[/tex] does not have a closed-form solution in terms of elementary functions. Therefore, we cannot determine if it converges or diverges analytically.

In this case, you may need to resort to numerical methods or approximations to evaluate the integral and determine its convergence or divergence.

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The cylindrical coordinate expression for f(x, y, z) = 7zex2 y2 z2 is: f(r, θ, z) = 7zexp(-2r^2) r^2 z^2.

Therefore, the cylindrical coordinate expression for f(x, y, z) = 7zex2 y2 z2 is: f(r, θ, z) = 7zexp(-2r^2) r^2 z^2

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Answer:

A

Step-by-step explanation:

The general Solution to the differential equation is

[tex]y(t) = y_h(t) + y_p(t) = c_1e^{7t} + c_2e^{-3t} + (1/25)e^{-3t}e^{4t}[/tex]

To find the general solution to the differential equation[tex]y'' - 4y' - 21y = e^{-3t}e^{4t}[/tex]

First, we find the homogeneous solution to the differential equation by solving the characteristic equation:

[tex]r^2 - 4r - 21 = 0[/tex]

The roots of this quadratic equation are:

r = 7, -3

So the homogeneous solution is:

[tex]y_h(t) = c_1e^{7t} + c_2e^{-3t}[/tex]

Next, we find a particular solution to the non-homogeneous equation. We can use the method of undetermined coefficients and assume a particular solution of the form:

[tex]y_p(t) = Ae^{-3t}e^{4t}[/tex]

Taking the derivatives of this function, we get:

[tex]y_p'(t) = (-3A + 4Ae^{-3t})e^{4t}y_p''(t) = (9A - 24Ae^{-3t} + 16Ae^{-6t})e^{4t}[/tex]

Substituting these derivatives and the assumed form of y_p(t) into the original differential equation, we get:

[tex](9A - 24Ae^{-3t} + 16Ae^{-6t})e^{4t} - 4(-3A + 4Ae^{-3t})e^{4t} - 21Ae^{-3t}e^{4t} = e^{-3t}e^{4t}[/tex]

Simplifying this equation, we get:

[tex](25A - 24)e^{4t} = e^{-3t}e^{4t}[/tex]

So, we have:

A = 1/25

Therefore, the particular solution is:

[tex]y_p(t) = (1/25)e^{-3t}e^{4t}[/tex]

Thus, the general solution to the differential equation is

[tex]y(t) = y_h(t) + y_p(t) = c_1e^{7t} + c_2e^{-3t} + (1/25)e^{-3t}e^{4t}[/tex]

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Answer:

To find the area of the trapezoid using decomposition, you can divide the trapezoid into a rectangle and two triangles. The area of the rectangle is x * h = 5 * 7 = 35 square units. The area of each triangle is (y - x) / 2 * h = (7 - 5) / 2 * 7 = 7 square units. So the total area of the trapezoid is 35 + 7 + 7 = 49 square units.

Step-by-step explanation:

there are 528 positive integers less than 1,000,000 that have exactly one digit equal to 9 and have a sum of digits equal to 13

To solve this problem, we need to consider the possible positions of the digit 9 in the number. Since we know that the number has exactly one digit equal to 9, there are six possible positions for the digit: in the ones place, in the tens place, in the hundreds place, in the thousands place, in the ten-thousands place, or in the hundred-thousands place.

Let's consider each of these cases separately:

Ones place: In this case, the remaining digits must sum up to 4, which is only possible if the number has the form 4XY, where X and Y are digits different from 9. There are 8 choices for X and 8 choices for Y, since they can be any digit from 0 to 8. Therefore, there are 8 * 8 = 64 numbers with 9 in the ones place and sum of digits equal to 13.

Tens place: In this case, the remaining digits must sum up to 3, which is only possible if the number has the form 3X9Y or 3Y9X. There are 8 choices for X and 7 choices for Y, since Y cannot be equal to X or 9. Therefore, there are 2 * 8 * 7 = 112 numbers with 9 in the tens place and sum of digits equal to 13.

Hundreds place: In this case, the remaining digits must sum up to 4, which is only possible if the number has the form 2X9Y or 2Y9X. There are 8 choices for X and 7 choices for Y, since Y cannot be equal to X or 9. Therefore, there are 2 * 8 * 7 = 112 numbers with 9 in the hundreds place and sum of digits equal to 13.

Thousands place: In this case, the remaining digits must sum up to 4, which is only possible if the number has the form 1X9Y or 1Y9X. There are 8 choices for X and 7 choices for Y, since Y cannot be equal to X or 9. Therefore, there are 2 * 8 * 7 = 112 numbers with 9 in the thousands place and sum of digits equal to 13.

Ten-thousands place: In this case, the remaining digits must sum up to 4, which is only possible if the number has the form X9Y0, where X and Y are digits different from 9. There are 8 choices for X and 8 choices for Y, since they can be any digit from 0 to 8. Therefore, there are 8 * 8 = 64 numbers with 9 in the ten-thousands place and sum of digits equal to 13.

Hundred-thousands place: In this case, the remaining digits must sum up to 4, which is only possible if the number has the form 9XY0, where X and Y are digits different from 9. There are 8 choices for X and 8 choices for Y, since they can be any digit from 0 to 8. Therefore, there are 8 * 8 = 64 numbers with 9 in the hundred-thousands place and sum of digits equal to 13.

To get the total number of numbers, we just add up the numbers from each case:

64 + 112 + 112 + 112 + 64 + 64 = 528

Therefore, there are 528 positive integers less than 1,000,000 that have exactly one digit equal to 9 and have a sum of digits equal to 13

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The  standard error of the sampling distribution for n = 52 is 0.0094, rounded to the nearest Hundredths.

The standard error of the sampling distribution of sample proportions for samples of size n = 32, 42, and n = 52, we first need to calculate the standard deviation (σ) of the population proportion, which is given by:

σ = sqrt [p * (1-p) / n]

Where p is the percentage of tight ends in the league who made a touchdown reception (p = 0.45) and n is the sample size.
For n = 32:

σ = sqrt [0.45 * (1-0.45) / 32] = 0.0805

The standard error (SE) of the sampling distribution of sample proportions is given by:

SE = σ / sqrt(n)

So, for n = 32:

SE = 0.0805 / sqrt(32) = 0.0143

The standard error for n = 32 is 0.0143, rounded to the nearest hundredths.

For n = 42:

σ = sqrt [0.45 * (1-0.45) / 42] = 0.0739

SE = σ / sqrt(n) = 0.0739 / sqrt(42) = 0.0114

Therefore, the standard error for n = 42 is 0.0114, rounded to the nearest hundredths.

For n = 52:

σ =[tex]\sqrt {0.45} * (1-0.45) / 52] = 0.0677SE = \sigma / \sqrt(n) \\= 0.0677 / \sqrt(52) = 0.0094[/tex]

Therefore, the standard error for n = 52 is 0.0094, rounded to the nearest hundredths.

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So the final expression for x(t) using only one cosine Function is:

x(t) = 2e^(-3t)cos(5t - (pi/2))

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Using this identity, we can rewrite the expression (-cos(5t) + 2sin(5t)) as follows:

(-cos(5t) + 2sin(5t)) = -sin(5t + (pi/2)) + sin(5t)

Notice that we've added pi/2 inside the sine function to shift the argument by 90 degrees. Now we can substitute this expression back into x(t):

x(t) = e^(-3t) * [ -sin(5t + (pi/2)) + sin(5t) ]

Using another trig identity, we can combine the two sine terms into one cosine term:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Using this identity, we get:

x(t) = e^(-3t) * [ 2cos(5t - (pi/2)) ]

So the final expression for x(t) using only one cosine function is:

x(t) = 2e^(-3t)cos(5t - (pi/2))

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The measure of angle R in kite RSTU is 45 degrees

To solve this problem, we can use the properties of a kite, which include two pairs of congruent adjacent sides and one pair of congruent opposite angles.

From the given diagram, we can see that the diagonals of kite RSTU intersect at point V, which is the midpoint of both diagonals. Additionally, we can see that sides RT and SU are congruent.

Since diagonals of a kite are perpendicular and bisect each other, we know that angles RVS and UVT are congruent and have a measure of 90 degrees each. Furthermore, since RT and SU are congruent, we know that angles RUT and STU are congruent and have a measure of 45 degrees each.

To find the measure of angle R, we can use the fact that the sum of the angles in a quadrilateral is equal to 360 degrees. Therefore, we have:

m∠R + m∠S + m∠T + m∠U = 360 degrees

Since the opposite angles in a kite are congruent, we know that m∠S = m∠U. Using this information along with the measures of angles RUT and STU, we can substitute in the following values:

m∠R + m∠S + 45 + 45 = 360 degrees

Simplifying the equation and substituting in m∠S for m∠U, we get:

2m∠R + 90 = 360 degrees

2m∠R = 270 degrees

m∠R = 135 degrees

However, this answer does not make sense, as the measure of angle R should be less than 90 degrees, since it is one of the angles of a kite.

We can see from the diagram that angle R is acute, so the measure of angle R must be less than 90 degrees. Therefore, the answer of 135 degrees is not valid.

Instead, we can use the fact that the two pairs of adjacent sides in a kite are congruent to determine the measure of angle R. Since RT and SU are congruent, we know that angles RST and TSU are congruent and have a measure of 67.5 degrees each.

Since the sum of the angles in a quadrilateral is 360 degrees, we have:

m∠R + m∠S + m∠T + m∠U = 360 degrees

Substituting in the measures of angles S and T, we get:

m∠R + 67.5 + 67.5 + 90 = 360 degrees

Simplifying the equation, we get:

m∠R = 135 - 90 = 45 degrees

Therefore, the measure of angle R in kite RSTU is 45 degrees

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complete question:

Quadrilateral RSTU is a kite. What is the measure of angle R?