Find the local and absolute maximum and minimum points in (x, y) format for the

function f(x) = 3/5x^5 - 9x^3 + 2 on the closed interval [-4,5]. Answer the following

questions.

a) Find all critical numbers (x- coordinates only)

b) Find the intervals on which the graph is increasing Mark critical numbers

Answer:

Step-by-step explanation:

To simplify the expression, we can combine the square roots and simplify the exponents.

Starting with the expression:

3√(16x^7) * 3√(12x^9)

Let's simplify each term separately:

Simplifying 3√(16x^7):

The index of the radical is 3, so we need to group the terms in sets of three. For the variable x, we have x^7, which can be grouped as x^6 * x.

Now, let's simplify the number inside the radical:

16 = 2^4, and we can rewrite it as (2^3) * 2 = 8 * 2.

So, 3√(16x^7) becomes:

3√(8 * 2 * x^6 * x) = 2 * x^2 * 3√(2x)

Simplifying 3√(12x^9):

Again, the index of the radical is 3, and we group the terms in sets of three. For the variable x, we have x^9, which can be grouped as x^6 * x^3.

Now, let's simplify the number inside the radical:

12 = 2^2 * 3.

So, 3√(12x^9) becomes:

3√(2^2 * 3 * x^6 * x^3) = 2 * x^2 * 3√(3x^3)

Now we can multiply the simplified terms together:

(2 * x^2 * 3√(2x)) * (2 * x^2 * 3√(3x^3))

Multiplying the coefficients: 2 * 2 * 3 = 12.

Multiplying the variables: x^2 * x^2 = x^4.

Now, let's combine the square roots:

3√(2x) * 3√(3x^3) = 3√(2x * 3x^3) = 3√(6x^4).

Therefore, the simplified expression is:

12x^4 * 3√(6x^4)

The daily cost of keeping a 50-watt bulb lit for 9 hours each day would be approximately $0.045 (or 4.5 cents).

To calculate the daily cost of keeping a 50-watt bulb lit for 9 hours each day, we need to consider the electricity cost per kilowatt-hour (kWh) and convert the wattage to kilowatts.

First, we convert the wattage of the bulb to kilowatts by dividing it by 1000:

50 watts = 50/1000 = 0.05 kilowatts.

Next, we calculate the total energy consumed by multiplying the power (in kilowatts) by the time (in hours):

Total energy consumed = 0.05 kW * 9 hours = 0.45 kilowatt-hours (kWh).

Now, we multiply the total energy consumed by the cost per kilowatt-hour to find the daily cost:

Daily cost = 0.45 kWh * $0.10/kWh = $0.045.

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We can solve for d = 150 / (tan(34) - tan(12)) Once we have the value of d, we can substitute it back into the equation h = d * tan(12) to find the height h of the smaller building.

To find the height h of the smaller building, we can use trigonometric ratios and the concept of angles of depression.

Let's denote the height of the smaller building as h. We are given that the observer in the larger building measures angles of depression of 12 degrees and 34 degrees to the top and bottom of the smaller building, respectively.

From the given information, we can form a right triangle with the vertical distance between the observer and the smaller building as the opposite side and the horizontal distance between the observer and the smaller building as the adjacent side.

Using trigonometric ratios, we can set up the following equations:

For the angle of depression of 12 degrees:

tan(12) = h / d

For the angle of depression of 34 degrees:

tan(34) = (h + 150) / d

Here, d represents the horizontal distance between the observer and the smaller building.

We can solve these two equations simultaneously to find the values of h and d.

From the equation for the angle of depression of 12 degrees, we can rewrite it as:

h = d * tan(12)

Substituting this expression for h in the equation for the angle of depression of 34 degrees, we get:

tan(34) = (d * tan(12) + 150) / d

Now, we can solve this equation for d. Rearranging the equation, we have:

d * tan(34) = d * tan(12) + 150

Simplifying further:

d * (tan(34) - tan(12)) = 150

Finally, we can solve for d:

d = 150 / (tan(34) - tan(12))

Once we have the value of d, we can substitute it back into the equation h = d * tan(12) to find the height h of the smaller building.

Note: To obtain an actual numerical value for h, we need the precise values of the tangent of 12 degrees and 34 degrees.

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Answer:  See the flowchart proof below.

Explanation:

The given info is indicated by the marked angles. We also use the reflexive property to say that BG = BG. After that we use the ASA (angle side angle) property to prove the triangles are congruent. The triangles are mirrored clones of one another. The mirror line is segment BG.

A flowchart proof that proves that the two triangles are congruent is shown in the image below.

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

In this context, we can prove that triangle BIG is congruent with triangle BAG by completing the two-column proof shown above with the following reasons:

Statements                                Reasons

∠IBG ≅ ∠ABG                              Given

∠IGB ≅ ∠AGB                              Given

BG ≅ BG                               Reflexive property

ΔBIG ≅ ΔBAG                       ASA Congruence

Based on the angle, side, angle (ASA) similarity theorem, we can logically deduce that triangle BIG and triangle BAG are both congruent.

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Answer:

211%

Step-by-step explanation:

percent = part/whole × 100%

percent = 19/9 × 100%

percent = 211%

The mean is the measure that gives the most accurate picture of the data's center. It is an essential measure of central tendency that represents the arithmetic average of a dataset.

It is calculated by summing up all the values in the dataset and dividing the sum by the total number of values. The mean is suitable for datasets that have a normal or symmetrical distribution.

The mean is highly sensitive to outliers, which can significantly influence the average value. When outliers are present, it is appropriate to use other measures of central tendency such as the median or mode to obtain an accurate picture of the data's center.

The median is the middle value in a dataset arranged in ascending or descending order. It is not affected by outliers and is suitable for datasets with skewed distributions.

The mode is the most frequent value in the dataset. It is suitable for categorical data but can also be used for continuous data.

In summary, the mean is the most accurate measure of central tendency, but its accuracy can be improved by using the median or mode in datasets with outliers or skewed distributions.

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The average rate of change of f(x) is less than average rate of change of g(x). Then the correct option is A.

A statement, principle, or policy that creates the link between two variables is known as a function. Functions are found all across mathematics and are required for the creation of complex relationships.

The function f(x) is given in the table, then the rate of change of the function f(x) will be

[tex]\text{Rate of change of f(x)} = \dfrac{(-2 + 4.5)}{(-2 + 3)}[/tex]

[tex]\text{Rate of change of f(x)} = 2.5[/tex]

If function g is a quadratic function that contains the points (-3, 5) and (0, 14).

Then the rate of change of the function g(x) will be

[tex]\text{Rate of change of g(x)} = \dfrac{(14 - 5)}{(0 + 3)}[/tex]

[tex]\text{Rate of change of g(x)} = \dfrac{9}{3}[/tex]

[tex]\text{Rate of change of g(x)} = 3[/tex]

Thus, the average rate of change of f(x) is less than average rate of change of g(x).

Then the correct option is A.

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Using the property of vertical angles, the value of x is 24 degrees

Vertical angles are a pair of non-adjacent angles formed by the intersection of two lines. They are opposite each other and share a common vertex, but they are not adjacent angles.

When two lines intersect, they form four angles at the point of intersection. The vertical angles are the angles that are directly across from each other and are formed by opposite pairs of intersecting lines. In other words, if you draw a line segment connecting the vertices of the vertical angles, it will divide the intersection into two pairs of congruent angles.

The property of vertical angles is that they have equal measures. This means that if one vertical angle measures a certain number of degrees, the other vertical angle will also measure the same number of degrees.

In this problem, to find the value of x, we have to use the property of vertical angles which states that vertical angles are equal to one another.

This implies that ∠ABC = DBE

3x - 14 = 2x + 10

Collect like terms

3x - 2x = 10 + 14

x = 24°

The answer is option D.

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The area of the garden is 135 square yards.

Let's imagine you have a rectangular garden measuring 15 yards in length and 9 yards in width.

You want to find the area of this garden, which represents the amount of space inside the garden.

To find the area of a rectangle, you multiply the length by the width.

In this case, the length is 15 yards and the width is 9 yards.

Area = Length × Width

Area = 15 yards × 9 yards

Area = 135 square yards

This means that the garden can hold 135 square yards of grass, flowers, or any other objects you place inside it.

It's worth noting that the unit of measurement for area is always squared, such as square yards in this example.

This is because area is a two-dimensional measurement, representing the space within a flat surface.

By using the formula to calculate the area of a rectangle, you can easily determine the amount of space enclosed by any rectangular area when given the length and width.

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Answer:

52500

Step-by-step explanation:

Let there be x employees in the previous year

Now, the company has 32% more employees whis is 69300

i.e.

[tex]x + \frac{32}{100} x = 69300\\\\ \implies\frac{132x}{100} = 69300 \\\\\implies 132x = 6930000\\\\\implies x = \frac{6930000}{132}\\[/tex]

⇒ x = 52500

There were 52500 employees in the previous year

An equation representing a line perpendicular to y = -6/5 [tex]\times[/tex] x - 7 would be y = 5/6 [tex]\times[/tex] x + c, where c is any constant.

To determine a line that is perpendicular to the given line y = -6/5 [tex]\times[/tex] x - 7, we need to consider the slope of the given line.

The given line has a slope of -6/5.

For two lines to be perpendicular, their slopes must be negative reciprocals of each other.

The negative reciprocal of -6/5 can be found by flipping the fraction and changing the sign, which gives us 5/6.

Therefore, the equation of a line perpendicular to y = -6/5 [tex]\times[/tex] x - 7 will have a slope of 5/6.

To find the equation of this perpendicular line, we can use the point-slope form of a line, using a known point on the line.

Let's assume the line passes through the point (x1, y1).

The equation of the perpendicular line can be written as: y - y1 = (5/6) [tex]\times[/tex] (x - x1).

Since we do not have a specific point given, we cannot determine the exact equation of the perpendicular line without additional information.

In summary, the equation of a line perpendicular to y = -6/5 [tex]\times[/tex] x - 7 will have a slope of 5/6, but the specific equation depends on the point it passes through.

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The group of individuals fitting a description is called option D: Population, this is because, in statistics, a population is seen as am entire group of individuals, items, or elements that tends to have or share a common characteristics.

The term "population" describes the complete group of people or things that you are interested in investigating. It is the group of individuals or thing(s) about which you are attempting to draw conclusions.

There are infinite and finite populations. A population with a set quantity of people or things is said to be finite. An endless population is one that has an infinite amount of people or things.

Therefore, the correct option is D

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See full text below

A group of individuals fitting a description is the _____

Which of the term below fit the description above.

A.census

B.sample

C.parameter

D.population

The actual length corresponding to the 3-inch length on the model is 10.5 feet.

Given that the scale of the model is 1 inch to 3.5 feet, we can use this information to find the actual length corresponding to a given length on the model.

Let's denote:

Model's length = 3 inches

Actual length = ?

According to the given scale, 1 inch on the model represents 3.5 feet in reality. We can set up a proportion to find the actual length:

(1 inch) / (3.5 feet) = (3 inches) / (x feet)

Cross-multiplying, we get:

1 inch * x feet = 3 inches * 3.5 feet

Simplifying the equation:

x feet = 10.5 feet

Therefore, the actual length corresponding to the 3-inch length on the model is 10.5 feet.

In summary, the actual length is 10.5 feet.

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Answer:540 degrees

The sum of the measures of the angles of a five-sided polygon (pentagon) is 540 degrees123. This can be calculated using the angle sum formula, S = (n − 2) × 180°, where n is the number of sides in the polygon12. Since a pentagon has five sides, the sum of the interior angles is (5 − 2) × 180° = 540°123.

Step-by-step explanation:

Answer:

540°

Step-by-step explanation:

Since a pentagon has n=5 sides, then we have:

[tex]180(n-2)=180(5-2)=180(3)=540^\circ[/tex]

The proportion by which the later temperature exceeds the earlier temperature is 3/5 or 3:5.

To determine the proportion by which one temperature exceeds the other, we need to calculate the difference between the two temperatures and express it as a proportion of the starting temperature.

First, calculate the temperature difference between 7:00 am and 3:00 pm:

temperature difference = 24°C - 15°C = 9°C

Now, express the temperature difference as a proportion of the starting temperature (15°C).

Proportion = Temperature difference / Starting temperature

Proportion = 9/15

Proportion = 3/5

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Question in English

In a city the temperature was measured at 7:00 am and the value was 15°C, then it was measured at 3:00 pm the value was 24°C. Starting from the level of measurement of this variable, in what proportion does one temperature exceed the other, analyze your answer.

The value of 408^5 is approximately 273,539,283,056,896. This is a large number that can be accurately calculated using a calculator or computer program.

To work out the value of 408^5, we need to perform the exponentiation calculation. In this case, 408 is the base, and 5 is the exponent.

Calculating large exponentiations like this can be time-consuming and prone to error if done manually. However, with the help of a calculator or computer, we can easily find the result.

Using a calculator or computer program, we can calculate 408^5 and obtain the value. The result is a very large number:

408^5 ≈ 273,539,283,056,896

Therefore, the value of 408^5 is approximately 273,539,283,056,896.

In decimal notation, the value is written as:

273,539,283,056,896

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Step-by-step explanation:

To solve this problem, we need to use the product rule of differentiation and some trigonometric identities. Let's start by finding the derivative of y with respect to x:

y = (sin 2x) √(3+2x)

Using the product rule, we get:

dy/dx = (sin 2x) d/dx(√(3+2x)) + (√(3+2x)) d/dx(sin 2x)

To find these derivatives, we need to use the chain rule and the derivative of sin 2x:

d/dx(√(3+2x)) = (1/2√(3+2x)) d/dx(3+2x) = (1/√(3+2x))

d/dx(sin 2x) = 2cos 2x

Substituting these values, we get:

dy/dx = (sin 2x) / √(3+2x) + 2cos 2x (√(3+2x))

Now, we need to simplify this expression to the desired form. To do that, we can use the trigonometric identity:

sin 2x = 2sin x cos x

Substituting this value, we get:

dy/dx = 2sin x cos x / √(3+2x) + 2cos 2x (√(3+2x))

Now, we can use the trigonometric identity:

cos 2x = 1 - 2sin^2 x

Substituting this value, we get:

dy/dx = 2sin x cos x / √(3+2x) + 2(1 - 2sin^2 x)(√(3+2x))

Simplifying further, we get:

dy/dx = (2cos x - 4cos x sin^2 x) / √(3+2x) + 2√(3+2x) - 4sin^2 x√(3+2x)

Now, we can see that this expression matches the desired form:

dy/dx = sin 2x + (4 + Bx)cos 2x / √(3+2x)

where A = -4 and B = -2. Therefore, we have shown that:

dy/dr = sin 2x + (4 - 2x)cos 2x / √(3+2x)

where A = -4 and B = -2.

Similarity statement and justification for two similar figures with given measurements. The given figures are AB = 40, AC = 50, and BC = 60. For the second figure, YX = 37.5, YZ = 30, and ZX = 45.

To write a similarity statement, we compare the ratios of corresponding sides of the two figures. So, we can compare AB/BC to YX/ZX and AC/BC to YZ/ZX.AB/BC = 40/60 = 2/3YX/ZX = 37.5/45 = 5/6AC/BC = 50/60 = 5/6YZ/ZX = 30/45 = 2/3Since the ratios of the corresponding sides of both figures are the same, we can say that the two figures are similar.

A similarity statement for these figures can be written as:ΔABC ~ ΔXYZThis statement indicates that the two triangles ABC and XYZ are similar. The symbol ~ is used to denote similarity.

The justification for this similarity statement is based on the fact that the ratios of the corresponding sides of the two figures are equal.

Therefore, by the definition of similarity, we can conclude that the two triangles are similar.

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Answer:

A 1/2

Step-by-step explanation:

was on my test trust me on this one

The solution to the system is (a) a = 4/5, b = 5, c = -4 and d =-4

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

The augmented matrix

Where, we have

[tex]\left[\begin{array}{cccc|c}1&0&0&0&4/5\\0&1&0&0&5\\0&0&1&0&-4\\0&0&0&1&-4\end{array}\right][/tex]

From the above, we have the diagonals to be 1

And other elements to be 0

This means that the equation has been solved

So, we have

a = 4/5, b = 5, c = -4 and d =-4

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Answer:

The range of this graph is (-4, 6], or

-4 < y < 6.

The range of the graph is [-4, 6].

In Mathematics and Geometry, a range is the set of all real numbers that connects with the elements of a domain.

This ultimately implies that, a range simply refers to the set of all possible output numerical values (real numbers), which are shown on the y-coordinate (y-axis) of a graph.

Based on the information provided in this scenario, the domain and range of the graph of this equation can be determined are as follows:

Domain = [1, 10] or 1 ≤ x < 10

Range = [-4, 6], or -4 < y ≤ 6.

Therefore, the range can be rewritten as [-4, 6].

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Answer:

Based on the values in the table, it can be observed that changing the radius of the sphere has a direct effect on the surface area of the sphere. As the radius increases, the surface area also increases.

Step-by-step explanation:

In general, multiplying the radius of a sphere by a factor of x will result in the surface area of the sphere being multiplied by a factor of x^2. This is because the surface area of a sphere is given by the formula:

Surface Area = 4πr^2

When the radius is multiplied by a factor of x, the new radius becomes xr. Substituting this into the formula, we get:

New Surface Area = 4π(xr)^2

= 4πx^2r^2

= x^2(4πr^2)

Therefore, the surface area of the sphere is multiplied by a factor of x^2 when the radius is multiplied by a factor of x. This relationship shows that the surface area increases at a faster rate than the radius.

The correct statement is C. The mean of East Hills HS is greater than the mean of Southview HS. So, option C is the correct answer.

To compare the centers of the distributions, we need to consider the measures of central tendency such as the median and the mean.

Looking at the given information about the class sizes at East Hills HS and Southview HS:

East Hills HS class sizes: 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44

Southview HS class sizes: 0, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44

Let's compare the measures of central tendency:

Median: The median is the middle value of a dataset when arranged in ascending or descending order. In this case, both datasets have an odd number of values, so the median is the middle value.

For East Hills HS: Median = 34

For Southview HS: Median = 34

Therefore, the medians of both distributions are the same.

Mean: The mean is calculated by summing all the values in the dataset and dividing by the total number of values.

For East Hills HS: Mean = (24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44) / 11 ≈ 35.727

For Southview HS: Mean = (0 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44) / 12 ≈ 33.667

Therefore, the mean of East Hills HS is greater than the mean of Southview HS.

Based on the comparisons, the correct statement is:

C. The mean of East Hills HS is greater than the mean of Southview HS.

So, option C is the correct answer.

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The question is asking for comparison of the centers and range of two data sets - the class sizes at two different high schools. Because no specific numeric data given for the mean and median of both school sizes, it's not possible to definitively choose an answer. Theoretically, if Southview has a higher concentration at larger class sizes, its median and mean might be larger.

This question refers to comparing the statistical centers (mean and median) and range of two data sets, which are the class sizes at East Hills High School and Southview High School. The centers and range of a data distribution tell us about the typical values, variety, and spread in the data set.

Without actual numbers (i.e., mean and median) for both the East Hills HS and Southview HS distributions, we can't definitively answer this question. However, let's assume that the representations provided suggest that Southview HS has a higher concentration of students in larger class sizes. In that case, the mean and median for Southview HS could be greater than those for East Hills HS, which would suggest option A and D. With regards to the range, if the minimum and maximum values of class sizes are the same in both schools, then the range, which is the difference between the maximum and minimum, would be the same. Therefore, option B wouldn't be accurate. However, as previously mentioned, this can only be suggestive as actual numbers are required for a definite answer.

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Answer:

The fish tank holds 40,000 ml of water.

Step-by-step explanation:

Volume = Length x Width x Height

Volume = 50 x 40 x 20

Volume = 40,000

a. For a score of X = 70, a standard deviation of σ = 4 would give a better grade.

b. For a score of X = 80, both standard deviations would give the same grade.

a. To determine which standard deviation would give a better grade for a score of X = 70, we can compare the z-scores associated with each standard deviation.

The z-score measures the number of standard deviations a given value is from the mean.

For σ = 4:

Z = (X - μ) / σ

Z = (70 - 78) / 4

Z = -2  

For σ = 8:

Z = (X - μ) / σ

Z = (70 - 78) / 8

Z = -1

The z-score for σ = 4 is -2, while the z-score for σ = 8 is -1. A higher z-score indicates a better grade since it represents a score that is further above the mean.

Therefore, in this case, a standard deviation of σ = 4 would give a better grade.

b. Similarly, for a score of X = 80:

For σ = 4:

Z = (X - μ) / σ

Z = (80 - 78) / 4

Z = 0.5

For σ = 8:

Z = (X - μ) / σ

Z = (80 - 78) / 8

Z = 0.25.

The z-score for σ = 4 is 0.5, while the z-score for σ = 8 is 0.25.

Again, a higher z-score indicates a better grade.

Therefore, in this case, a standard deviation of σ = 4 would give a better grade.

In both scenarios, a standard deviation of σ = 4 would result in a better grade compared to σ = 8.

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The cosine function repeats its pattern every 2π radians (or 360 degrees), we can say that the period of y = cos(x) is 2π.

The period of the function y = cos(x) is 2π.

To understand the period of the cosine function, we need to examine its graph. The cosine function is a periodic function that oscillates between -1 and 1 as x varies. It repeats its pattern over regular intervals.

The cosine function completes one full cycle from 0 to 2π radians (or 0 to 360 degrees). This means that within this interval, the cosine function goes through one complete oscillation, starting from its maximum value of 1, then going through its minimum value of -1, and returning back to 1.

Since the cosine function repeats its pattern every 2π radians (or 360 degrees), we can say that the period of y = cos(x) is 2π.

This means that for any value of x, the value of cos(x) will repeat after an interval of 2π.

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IQR, because Sunny Town is symmetric should be used for both sets of data to determine the location with the most consistent temperature?(option a).

1. The question asks for the measure of variability that should be used to determine the location with the most consistent temperature when comparing the data from two different locations.

2. The first option, A, suggests using the Interquartile Range (IQR) because Sunny Town is symmetric. This means that the data in Sunny Town is evenly distributed around the median, indicating consistency in temperatures.

3. The second option, B, proposes using the IQR because Beach Town is skewed. Skewness implies an asymmetrical distribution, which may indicate less consistency in temperatures.

4. The third option, C, suggests using the Range because Sunny Town is skewed. Skewed data in Sunny Town might imply a larger spread and less consistency in temperatures.

5. The fourth option, D, recommends using the Range because Beach Town is symmetric. However, symmetric data indicates consistency, making the Range less suitable as a measure of variability.

6. Considering the explanations for each option, the best choice is A, IQR, because Sunny Town is symmetric. The symmetric distribution suggests that the temperatures in Sunny Town are consistent and evenly distributed around the median.

7. Therefore, the measure of variability that should be used for both sets of data to determine the location with the most consistent temperature is the IQR, as indicated by option A.

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