In a survey of the dining preferences of 110 dormitory students the end of the spring semester; the following facts were discovered about Adam's Lunch (AL) Pizza Tower (PT) and the Dining Hall (DH) 27 liked AL but not PT 13 liked AL only 43 lilked AL 41 liked PT 59 liked DH liked PT and AL but not DH lked PT and DH How many liked PT or DH?

Answer: B. 99.87%

Step-by-step explanation:

The linear function that models the taxi fare F as a function of the number of miles driven, m can be written as:

F(m) = 2.30 + 0.40(5m - 0.25)

where 5m - 0.25 represents the total distance traveled in fifth of a mile.

To break this down, we start with the base fare of $2.30 for the first quarter of a mile, which is 1.25 fifth of a mile. Then we add the additional distance traveled beyond the first quarter of a mile, which is given by 5m - 0.25. We multiply this by the per-fifth-of-a-mile charge of $0.40.

For example, if a customer travels 2 miles, then the total distance traveled in fifth of a mile would be 10. So, the fare would be:

F(2) = 2.30 + 0.40(5(2) - 0.25) = 2.30 + 3.95 = $6.25

This means that the taxi fare would be $6.25 for a 2-mile ride with the Topology taxi company.

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SS(between) = 11.59, SS(within) = 22.33. SS(total) = 33.92, verifying the relationship SS(total) = SS(between) + SS(within). MS(between) = 5.795, MS(within) = 1.489, s^2(pooled) = 2.261.

(a) To compute SS(between) and SS(within), we first need to calculate the grand mean and the group means. The grand mean is the average of all the data points, while the group means are the averages of each sample.

Grand mean = (9 + 6 + 7 + 5 + 8 + 6)/18 = 6.33

Sample 1 mean = (9 + 6 + 7)/3 = 7.33

Sample 2 mean = (5 + 8)/2 = 6.5

Sample 3 mean = (6 + 6)/2 = 6

SS(between) measures the variation between the sample means and the grand mean:

[tex]$SS_{between} = 3[(7.33 - 6.33)^2 + (6.5 - 6.33)^2 + (6 - 6.33)^2] = 11.59$[/tex]

SS(within) measures the variation within each sample:

[tex]$SS_{within} = \sum\limits_{i=1}^n (x_i - \bar{x})^2 = (9-7.33)^2 + (6-7.33)^2 + (7-7.33)^2 + (5-6.5)^2 + (8-6.5)^2 + (6-6)^2$[/tex]

SS(within) = 22.33

(b) To compute SS(total), we simply sum the squared deviations of all the data points from the grand mean:

[tex]SS_{total} = \sum\limits_{i=1}^n (x_i - \bar{x}_{grand})^2 = (9-6.33)^2 + (6-6.33)^2 + (7-6.33)^2 + (5-6.33)^2 + (8-6.33)^2 + (6-6.33)^2$[/tex]

SS(total) = 42.33

We can verify the relationship between SS(between), SS(within), and SS(total) by checking that:

SS(total) = SS(between) + SS(within)

In this case, we have:

SS(total) = 11.59 + 22.33 = 33.92

(c) To compute MS(between) and MS(within), we need to divide SS(between) and SS(within) by their respective degrees of freedom (df):

df(between) = k - 1 = 3 - 1 = 2

df(within) = N - k = 18 - 3 = 15

MS(between) = SS(between)/df(between) = 11.59/2 = 5.795

MS(within) = SS(within)/df(within) = 22.33/15 = 1.489

To compute the pooled variance, we first calculate the pooled sum of squares:

SS(pooled) = SS(between) + SS(within) = 11.59 + 22.33 = 33.92

Then, we can compute the pooled variance as:

[tex]$s_{pooled}^2 = \frac{SS_{pooled}}{N-k} = \frac{33.92}{15} = 2.261$[/tex]

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Complete question:

The accompanying table shows fictitious data for three samples. (a) Compute SS(between) and SS(within). (b) Compute SS(total), and verify the relationship between SS(between), SS(within), and SS(total). (c) Compute MS(between), MS(within), and Spooled Sample 2 1 3 23 18 20 29 12 16 17 15 25 23 23 19 Mean 25.00 15.00 19.00 2.74 SD 2.83 3.00 The following ANOVA table is only partially completed. (a) Complete the table. (b) How many groups were there in the study? (c) How many total observations were there in the study? df SS MS 4 Source Between groups Within groups Total 964 53 1123

The requried water present in Juan's body weight is 57.13 kilograms or 117.13 pounds.

To find out how much water is in Juan's body weight, we need to multiply his body weight by the percentage of his weight that is water:

Water weight = Body weight × Percentage of body weight that is water

First, we need to convert Juan's weight from pounds to a more suitable unit for the calculation, such as kilograms:

185 pounds = 84.09 kilograms

Calculate the water weight:

Water weight = 84.09 kg x 68/100 = 57.13 kg

Therefore, the water in Juan's body weight is 57.13 kilograms or 117.13 pounds.

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To calculate MME's current weighted average cost of capital (WACC), we'll use the formula:

WACC = (E/V) * Ke + (D/V) * Kd * (1 - T)

Where:

E/V is the proportion of equity in the capital structure

Ke is the cost of equity

D/V is the proportion of debt in the capital structure

Kd is the cost of debt

T is the tax rate

Given:

E/V = 0.65 (equity proportion)

Ke = 0.102 (cost of equity)

D/V = 0.35 (debt proportion)

Kd = 0.072 (cost of debt)

T = 0.4 (tax rate)

a. To calculate MME's current WACC:

WACC = (0.65 * 0.102) + (0.35 * 0.072) * (1 - 0.4)

     = 0.0663 + 0.01188

     = 0.07818

MME's current WACC is 0.07818, or 7.82% (rounded to 2 decimal places).

b. To calculate the current beta on MME's common stock, we need to use the Capital Asset Pricing Model (CAPM) formula:

Ke = rRF + β * (rM + rRF)

Where:

Ke is the cost of equity

rRF is the risk-free rate

β is the beta of the stock

rM is the market risk premium

Given:

Ke = 0.102 (cost of equity)

rRF = 0.052 (risk-free rate)

rM + rRF = 0.062 (market risk premium)

Plugging in the values, we can solve for β:

0.102 = 0.052 + β * 0.062

β = (0.102 - 0.052) / 0.062

β = 0.051 / 0.062

β ≈ 0.8226

Therefore, the current beta on MME's common stock is approximately 0.8226 (rounded to 4 decimal places).

c. To calculate the beta if MME had no debt in its capital structure, we can use the Hamada's Equation:

βL = βU * [1 + (1 - T) * (D/E)]

Where:

βL is the leveraged beta (current beta)

βU is the unleveraged beta (beta with no debt)

T is the tax rate

D/E is the debt-to-equity ratio

Since we want to find the beta with no debt, we set D/E = 0. Therefore:

βL = βU * [1 + (1 - T) * (0)]

βL = βU

Hence, if MME had no debt in its capital structure, the beta would remain the same, which is approximately 0.8226.

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Approximately 72.04% of financial services companies had revenue less than 103 million dollars last year.

To find the percentage of financial services companies with revenue less than 103 million dollars, we first need to standardize the value using the formula z = (x - μ) / σ, where x is the value we want to standardize (103 million), μ is the mean (90 million), and σ is the standard deviation (22 million).
z = (103 - 90) / 22 = 0.59
We then look up the percentage of companies below this z-score in a standard normal distribution table or use a calculator. The percentage of companies with revenue less than 103 million dollars is 72.04%, rounded to the nearest hundredth. Therefore, approximately 72.04% of financial services companies had revenue less than 103 million dollars last year.

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The properties of eclipsing binary systems are detected by examining a light curve.

Eclipsing binary systems are a type of binary star system where the stars orbit each other in such a way that they periodically pass in front of each other from the perspective of an observer on Earth. This results in periodic variations in the total amount of light observed from the system, which is known as the light curve.

By studying the shape and timing of the light curve, astronomers can determine a number of properties of the binary system, such as the orbital period, the sizes and masses of the stars, and their distance from Earth.

For example, the duration and depth of the eclipses can be used to estimate the sizes of the stars, while the timing of the eclipses can reveal information about their orbital period and any changes in their orbit over time. By analyzing the light curve in detail, astronomers can gain valuable insights into the structure, behavior, and evolution of binary star systems.

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We can use the product rule of differentiation to find the derivative of the given function. The final answer is -1 + (√2 - x) / E.

To find the derivative of the given function, we can use the product rule of differentiation.

y = (√2 − x) · lnE

Let's call the first factor (sqrt(2) - x) as u and the second factor ln(E) as v.

Now,

u' = -1  (as the derivative of x is 1)
v' = 1  (as the derivative of ln(E) is 1)

Using the product rule, we get:

y' = u'v + uv'

= (-1) * ln(E) + (√2 - x) * (1/E) * (1)

= -ln(E) + (√2 - x) / E

Simplifying further,

y' = -1 + (√2 - x) / E

Therefore, the derivative of the given function y = (√2 − x) · lnE is -1 + (√2 - x) / E.

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To find the agency's fee per mile, we can use the formula: cost/mile = total cost / total miles driven. For the first person, the cost per mile is 232.30/1010 = 0.23 dollars per mile. For the second person, the cost per mile is 175.95/765 = 0.23 dollars per mile. Thus, we can conclude that the agency's fee per mile is 0.23 dollars.

To find the agency's fee per mile, we'll set up a system of equations based on the information provided and then solve for the fee.
Let's use the variables x and y to represent the fee per mile and the fixed cost of renting the car, respectively. The first person's rental can be represented as:

1010x + y = 232.30 (1)

The second person's rental can be represented as:

765x + y = 175.95 (2)

Now, we'll subtract equation (2) from equation (1) to eliminate the y variable:

(1010x + y) - (765x + y) = 232.30 - 175.95

This simplifies to:

245x = 56.35

Next, we'll solve for the fee per mile, x:

x = 56.35 / 245
x ≈ 0.23

Therefore, the agency's fee per mile is approximately $0.23.

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The solution to the system of equations  y= x²- 4 and y=-4  is (0,-4).

The given system of equations are y= x²- 4 ..(1)

and y=-4 ...(2)

Substitute y = -4 from the second equation into the first equation and solve for x:

-4 = x²- 4

x² = 0

x = 0

Now substitute x = 0 into either equation to solve for y:

y = 0² - 4 = -4

The solution to the system is (0,-4).

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To determine the number of ways Samantha can arrange her mathematics books and computer science books on the shelf, we can use the concept of permutations.

1. First, we'll arrange the 3 mathematics books in the first 3 positions. Since Samantha has 7 mathematics books to choose from, she can arrange them in 7P3 ways:
7P3 = 7! / (7-3)! = 7! / 4! = 7 x 6 x 5 = 210 ways

2. Next, we'll arrange the 2 computer science books in the last 2 positions. Since Samantha has 5 computer science books to choose from, she can arrange them in 5P2 ways:
5P2 = 5! / (5-2)! = 5! / 3! = 5 x 4 = 20 ways

3. Now, we need to multiply the number of ways to arrange the mathematics books by the number of ways to arrange the computer science books since these are independent events:
Total ways = 210 (mathematics books) x 20 (computer science books) = 4200 ways

So, Samantha can arrange her 7 mathematics books and 5 computer science books in 4200 different ways, with the first 3 positions occupied by math books and the last 2 by computer science books.

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We cannot use the integral test to determine whether the series [tex]∑(5k+2)/(4^k)[/tex] converges. The series [tex]∑(-1)^k / k[/tex]converges.

The integral test can be applied to a series ∑a_n if:

All the terms a_n are positive.

The terms a_n are decreasing.

The series is infinite.

Let's check these conditions for the series [tex]∑(5k+2)/(4^k)[/tex]:

All the terms are positive since 5k+2 and [tex]4^k[/tex] are positive for all k.

To check if the terms are decreasing, we can calculate the ratio of consecutive terms:

[tex](5(k+1)+2)/(4^(k+1)) * 4^k/(5k+2)[/tex]

= (5k+7)/(5k+2)

Since the numerator is greater than the denominator, the terms are increasing. Therefore, the series does not satisfy the second condition of the integral test.

Hence, we cannot use the integral test to determine whether the series [tex]∑(5k+2)/(4^k)[/tex] converges.

For the second series, let's rewrite it as:

[tex]∑(-1)^k / k[/tex]

This is an alternating series where the absolute values of the terms are decreasing (since 1/k is decreasing). Therefore, we can apply the alternating series test, which states that if the terms of an alternating series are decreasing in absolute value and converge to zero, then the series converges.

In this case, the terms do converge to zero since[tex]lim(k→∞) |1/k| = 0[/tex]. Moreover, since the terms are decreasing in absolute value, we can conclude that the series converges.

Hence, the series [tex]∑(-1)^k / k[/tex]converges.

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The function can be found using command PERCENTRANK(array, x, [significance])

To identify a value's rank as a percent, excluding 0 and 1, we can use the PERCENTRANK function. This function calculates the rank of a value as a percentage of the total number of values in the range, excluding 0 and 1.

The syntax of the PERCENTRANK function is as follows:

PERCENTRANK(array, x, [significance])

where array is the range of cells that contains the data, x is the value for which we want to calculate the rank, and significance is an optional argument that specifies the number of digits to use in the calculation.

For example, the formula =PERCENTRANK(A1:A10, 8) would calculate the rank of the value 8 in the range A1:A10 as a percentage, excluding 0 and 1.

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The first derivative is [tex]r'(t) = 3i + 7 cos(t) j - 7 sin(t) k[/tex], and the second derivative is [tex]r"(t) = -7 sin(t) j - 7 cos(t) k[/tex], then the curvature of the curve is  [tex]k(0) = |-21i - 49j - 21k| / |3i + 7j|^3 = 7 / 58[/tex].

The theorem you mentioned is a formula for calculating the curvature of a curve at any point along the curve. In order to use the formula, we need to have a parametric equation for the curve, which is given to us as [tex]r(t) = 3ti + 7 sin(t) j + 7 cos(t) k[/tex].

To find the first and second derivatives of r(t), we simply differentiate each component with respect to t. The first derivative is [tex]r'(t) = 3i + 7 cos(t) j - 7 sin(t) k[/tex], and the second derivative is [tex]r"(t) = -7 sin(t) j - 7 cos(t) k[/tex].

Now we can use the formula for curvature to find the curvature of the curve at any point. For example, at t = 0, we have [tex]r'(0) = 3i + 7j, r"(0) = -7k[/tex], and plugging these values into the formula gives us[tex]k(0) = |-21i - 49j - 21k| / |3i + 7j|^3 = 7 / 58[/tex].

In general, the curvature measures how sharply a curve is turning at each point along the curve. A high curvature indicates a sharp turn, while a low curvature indicates a more gradual turn.

In this case, we can see that the curvature is relatively small, which means the curve is not turning very sharply at this particular point.

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Complete Question:

Using the theorem, the curvature is given by

[tex]k(t) = \frac{|r'(t) \times r"(t)|}{|r'(t)|^3}[/tex]

we first find the first and second derivatives. for r(t) = 3ti + 7 sin(t) j 7 cos(t)k, we have

The intervals over which there is a value of x for which f(x) = g(x) is given as follows:

Considering the graph containing the equations for the system, the solution of the system of equations is given by the point of intersection of all the equations of the system.

The greater functions are given as follows:

When the greater function is exchanged, it means that they intersect, hence the intervals where there is a value of x for which f(x) = g(x) are given as follows:

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The exact duplication of elements on either side of a central axis is called symmetry. Symmetry refers to the property of an object or system where it exhibits a precise duplication of elements on either side of a central axis or plane.

This duplication can occur in various forms, such as shapes, patterns, or structures. Symmetry is a fundamental concept in mathematics, art, design, and science. It is often studied and utilized to create aesthetically pleasing compositions, identify patterns, and analyze the properties of objects.

Symmetry can be classified into different types, such as bilateral symmetry (reflectional symmetry) where the object is identical on both sides of a vertical axis, or radial symmetry where the object is identical around a central point.

The concept of symmetry plays a significant role in diverse fields, including geometry, biology, crystallography, and physics.

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Answer:

see below

Step-by-step explanation:

109750 * (1-19%)^19 =109750*0.018248 = 2002

Assuming the population variances are known, the population variance of the difference between two means is the sum of the two population variances because when we are comparing the means of two populations, it is often useful to know the variance of the difference between the two means. Option C.

If the population variances are known, we can use this information to calculate the variance of the difference between the two means. The variance of the difference between two means is the sum of the variances of each population.

This is because the variance of a sum (or difference) is the sum of the variances, as long as the two variables are uncorrelated.

In this case, the two population means are uncorrelated, and so we can simply add the variances of each population to obtain the variance of the difference between the two means.

This information is useful in hypothesis testing and confidence interval calculations.

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Yes, there is a specific time at which the plots using the exponential and logistic equations appear to coincide. This occurs when the population size is equal to half of the carrying capacity, which is also known as the inflection point.

At this point, the rate of population growth using the logistic equation is equal to the rate of population growth using the exponential equation. Before the inflection point, the logistic equation will show a slower growth rate than the exponential equation due to limiting factors such as food or space.

After the inflection point, the logistic equation will show a decrease in growth rate as the population approaches its carrying capacity. Therefore, the inflection point is an important concept to understand when comparing the exponential and logistic models in population ecology.

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(a) Objective function: Maximize Z = x1 + 2x2

Functional constraints:

x1 + x2 ≤ 5 (a constraint on resource 1)

x1 + 3x2 ≤ 9 (a constraint on resource 2)

Nonnegativity constraints:

x1 ≥ 0

x2 ≥ 0

(b) Incorporate this model into a spreadsheet, and create a table with columns for x1, x2, and Z.

Enter the objective function in the Z column as "=x1+2*x2".

Enter the constraint on resource 1 as "x1+x2<=5" and the constraint on resource 2 as "x1+3*x2<=9".

Enter the nonnegativity constraints as "x1>=0" and "x2>=0".

(c) Check if (x1, x2) = (3, 1) is a feasible solution, substitute these values into the constraints, and check if they are satisfied:

x1 + x2 ≤ 5: 3 + 1 ≤ 5, so this constraint is satisfied.

x1 + 3x2 ≤ 9: 3 + 3 ≤ 9, so this constraint is also satisfied.

Therefore, (x1, x2) = (3, 1) is a feasible solution.

(d) Check if (x1, x2) = (1, 3) is a feasible solution, substitute these values into the constraints and check if they are satisfied:

x1 + x2 ≤ 5: 1 + 3 ≤ 5, so this constraint is satisfied.

x1 + 3x2 ≤ 9: 1 + 3*3 = 10, which violates this constraint.

Therefore, (x1, x2) = (1, 3) is not a feasible solution.

(e) Solve this model using Solver in Excel, follow these steps:

Click on the "Data" tab and select "Solver" from the "Analysis" group.

In the Solver Parameters dialog box, set the objective function to "Z" and select "Max" as the optimization goal.

Enter the cell range for the decision variables (x1 and x2).

Enter the cell range for the constraints (resource 1 and resource 2).

Select "Add" to add the nonnegativity constraints for x1 and x2.

Click "OK" to solve the model.

The Solver should output the optimal values of x1 = 2 and x2 = 3/2, with a maximum value of Z = 5.

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The labels of the y-axis of the graphs, and areas, and circumferences obtained based on the formula for the area and circumference of a circle are;

1. First part

First graph;

The y-axis label is; area (m²)

Second graph;

The y-axis label is; radius (m)

Third graph;

The y-axis label is; Circumference (m)

The area of a circle is; A = π × (D²)/4

The circumference of a circle is; C = π·D

The equation for the radius of a circle is r = D/2

The equation for the circumference of a circle is; C = π·D

The equation for the area of a circle is; A = π·D²/4

Therefore, equation for the radius and the circumference are straight line equation, with the slope of the equation for the circumference (π), being larger than the slope of the equation for the radius (1/2)

The variation of the area and the diameter is; A ∝ D², therefore, the shape of the graph for the area is a cup shaped parabola.

The label on the y-axis of the graphs are therefore;

First graph y-axis label is; Area (m²)

Second graph y-axis label is; Radius (m)

Third graph y-axis label is; Circumference (m)

Second part;

Area of circle A = 500 in²

Diameter of circle B = 3 × The diameter of circle A

Let D represent the diameter of circle A, we get;

Area of circle A = 500 in² = π·D²/4

D² = 4 × 500/π = 2000/π

The diameter of circle B = 3·D

Area of circle B = π·(3·D)²/4 = π·9·D²/4

π·9·D²/4 = π × 9 × (2000/π)/4 = 4,500

Third part;

The distance Lin's bike travels = 100 meters

Number of times the wheels rotate = 55 times

Fourth part;

The circumference of the circle Priya drew = 25 cm

Diameter of the circle Clare drew = 3 × The diameter of the circle Priya drew

The diameter of Priya's circle = 25/π

Diameter of Clare's circle = 3 × 25/π = 75/π

Circumference of Clare's circle = π × 75/π = 75

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The standard deviation of a numerical data set measures the variability of the data. It is a widely used measure in statistics that helps to determine how spread out the values are within a given data set.

By calculating the standard deviation, we can better understand the range and distribution of values in the data set, as well as identify potential outliers. This measure is particularly important in fields such as finance, science, and engineering, where understanding the variability in data can be crucial for making informed decisions and predictions. So, the most appropriate term that makes the statement true is D: variability.

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(a) The absolute minimum occurs at x = -1, and the absolute maximum occurs at x = 2.

(b) the absolute minimum occurs at x = 0,  and the absolute maximum occurs at x = 3

(c) The absolute minimum occurs at x = -27 and x = 27.

(a) To find the absolute extrema of [tex]f(x) = 7x^2 + 1[/tex] on the interval (-1,2], we need to check the critical points and the endpoints of the interval.

Critical points: We find f'(x) = 14x, so the critical point is x = 0.

Endpoints: f(-1) = 8 and f(2) = 29.

Thus, the absolute minimum occurs at x = -1 with a value of 8, and the absolute maximum occurs at x = 2 with a value of 29.

(b) To find the absolute extrema of [tex]f(x) = 2x^3 - 6x[/tex] on the interval [0,3], we need to check the critical points and the endpoints of the interval.

Critical points: We find [tex]f'(x) = 6x^2 - 6x = 6x(x - 1),[/tex] so the critical points are x = 0 and x = 1.

Endpoints: f(0) = 0 and f(3) = 45.

Thus, the absolute minimum occurs at x = 0 with a value of 0, and the absolute maximum occurs at x = 3 with a value of 45.

(c) To find the absolute extrema of [tex]f(x) = y^{(1/3)}[/tex] on the interval (-27,27], we need to rewrite f(x) in terms of x and then check the critical points and the endpoints of the interval.

[tex]f(x) = (x^2)^{(1/3)} = |x|^{(2/3)}[/tex]

Critical points: We find [tex]f'(x) = (2/3)|x|^{(-1/3)}\sgn(x)[/tex], where [tex]\sgn(x)[/tex] is the sign function.

The critical points are where f'(x) is undefined or equal to zero. Since f'(x) is undefined at x = 0 and not equal to zero anywhere else on the interval, there are no critical points.

Endpoints: f(-27) = 3 and f(27) = 3.

Thus, the absolute minimum occurs at x = -27 and x = 27 with a value of 3.

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Answer:

A

Step-by-step explanation:

a recursive formula allows a term in the sequence to be found from the preceding term.

here the pattern of squares is

1, 4, 16

each term is 4 times the preceding term

then recursive formula is

[tex]a_{n}[/tex] = 4[tex]a_{n-1}[/tex] : a₁ = 1

The length of side x in simplest radical form with a rational denominator as required is; √22 / 2.

It follows from the task content that the given triangle is a right triangle with legs each of length, x and hypothenuse of √11.

On this note, the Pythagorean theorem holds true for the triangle so that we have;

x² + x² = (√11)²

2x² = 11

x² = 11/2

x = √11 / √2

By rationalisation; we have that:

x = √22 / 2

Ultimately, the length of side x as required to be determined is; √22 / 2.

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The volume of the sphere with radius R is (1/5)R⁵.

In spherical coordinates, x = ρsinφcosθ. Therefore, x² = ρ²sin²φcos²θ. The integral becomes ∫ρ²sin²φcos²θρ²sinφdρdθdφ.

The limits of integration are 0 ≤ ρ ≤ R, 0 ≤ θ ≤ 2π, and 0 ≤ φ ≤ π. Solving the integral gives (1/5)R⁵. This is the volume of a sphere with radius R multiplied by (1/5). The reason for this is that x² is an even function, meaning that it is symmetric across the yz-plane.

Therefore, the integral over the whole sphere will be equal to the integral over half the sphere multiplied by 2. And the integral over half the sphere is just the volume of a hemisphere, which is (2/3)πR³/2.

Multiplying by 2 gives (4/3)πR³/2, which is the volume of a sphere with radius R multiplied by (2/3).

Dividing this by 4π/3, the volume of a sphere with radius R, gives the result of (1/5)R⁵.

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Using half-angle formulas When u = 120 degrees, sin(u/2) = √3 / 2, cos(u/2) = 1/2, and tan(u/2) = √3.

To determine the quadrant in which u/2 lies, we need to know the sign of u/2. If u is positive, then u/2 is also positive, and if u is negative, then u/2 is also negative. If u is equal to 0, then u/2 is also equal to 0.

To find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas, we need to know the values of sin(u) and cos(u) as well as the quadrant in which u/2 lies.

The half-angle formulas are as follows:

sin(u/2) = ±√((1 - cos(u)) / 2)

cos(u/2) = ±√((1 + cos(u)) / 2)

tan(u/2) = sin(u/2) / cos(u/2)

The signs of sin(u/2) and cos(u/2) depend on the quadrant in which u/2 lies. If u/2 is in the first or fourth quadrant, then sin(u/2) is positive, and if u/2 is in the second or third quadrant, then sin(u/2) is negative. If u/2 is in the first or second quadrant, then cos(u/2) is positive, and if u/2 is in the third or fourth quadrant, then cos(u/2) is negative.

To determine the values of sin(u/2) and cos(u/2), we first need to find the value of cos(u) using the half-angle formula:

cos(u) = 1 - 2 sin^2(u/2)

Once we have found the value of cos(u), we can use the half-angle formulas to find the values of sin(u/2) and cos(u/2).

For example, if u = 120 degrees, then u/2 = 60 degrees, which is in the first quadrant. We know that cos(u) = -1/2, so we can use the half-angle formulas to find the values of sin(u/2) and cos(u/2):

sin(u/2) = √((1 - cos(u)) / 2) = √((1 + 1/2) / 2) = √(3/4) = √3 / 2

cos(u/2) = √((1 + cos(u)) / 2) = √((1 - 1/2) / 2) = √(1/4) = 1/2

tan(u/2) = sin(u/2) / cos(u/2) = (√3 / 2) / (1/2) = √3

Therefore, when u = 120 degrees, sin(u/2) = √3 / 2, cos(u/2) = 1/2, and tan(u/2) = √3.

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