Evaluate the given integral by changing to polar coordinates.

∫∫x dA , where D is the region in the first quadrant that lies between the circles x^2 + y^2 = 16 and x^2 + y^2 = 4x

The population after 1.5 years will be 360640.9.

Given that, a town's population was 345,000 in 1996. Its population increased by 3% each year.

The exponential growth =

A = P(1+r)ⁿ

A = final amount, P = initial amount, r = rate and n = time.

A = 345000(1+0.03)ⁿ

A = 345000(1.03)ⁿ

There is a growth factor of 1.03.

For n = 1.5

[tex]A = 345000(1.03)^{1.5[/tex]

A = 360640.9

Hence, the population after 1.5 years will be 360640.9.

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There are default tab stops every one inch on the horizontal ruler is true statement.

In word processing software, there are usually default tab stops set every one inch (or 2.54 cm) on the horizontal ruler. These tab stops are the default positions at which the insertion point will stop when the tab key is pressed.

The purpose of these tab stops is to make it easier to align text in columns or tables. By default, there are left-aligned, centered, right-aligned, decimal-aligned, and bar-aligned tab stops. Users can also add custom tab stops at specific positions on the ruler to suit their formatting needs.

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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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[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=\sqrt{a^2 + o^2} \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{2}\\ o=\stackrel{opposite}{8} \end{cases} \\\\\\ c=\sqrt{ 2^2 + 8^2}\implies c=\sqrt{ 4 + 64 } \implies c=\sqrt{ 68 }\implies c\approx 8.2[/tex]

Answer:

8.3

Step-by-step explanation:

The equation to find the length of the hypotenuse from the legs is a² + b² = c². This means that you will need to square the lengths of the legs and then add them together to get the length of the hypotenuse squared. The lengths of the legs of this triangle are 2 and 8. 2 squared is 4, and 8 squared is 64. Add 64 + 4 to get 68. Now we know that 68 equals the hypotenuse squared. The square root of 68, 8.246, is the measurement of the hypotenuse. Now you just need to round up the number to the nearest tenth to get 8.3.

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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Answer:

see below

Step-by-step explanation:

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

(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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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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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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Answer: B. 99.87%

Step-by-step explanation:

Answer:

There can be no answer, as you did not provide the box plot to compare the data.

The probability that the child must wait at least 8 minutes for the bus on a given morning is 0.2636

To determine the probability that a child must wait at least 8 minutes for the bus on a given morning, given that the waiting time is exponentially distributed with a mean of 6 minutes, we'll use the exponential distribution formula:

[tex]P(T > t) = e^{(-t/μ)}[/tex]

where T is the waiting time, t is the specific time we are interested in (8 minutes in this case), μ is the mean waiting time (6 minutes), and e is the base of the natural logarithm (approximately 2.71828).

Step 1: Plug in the values into the formula:

[tex]P(T > 8) = e^{(-8/6)}[/tex]

Step 2: Simplify the exponent:

[tex]P(T > 8) = e^{(-4/3)}[/tex]

Step 3: Calculate the probability using the value of e:

[tex]P(T > 8) ≈ 2.71828^{(-4/3)} ≈ 0.2636[/tex]

Therefore, the probability that the child must wait at least 8 minutes for the bus on a given morning is approximately 0.2636, which corresponds to option (a).

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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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1. The rate of change is -1 and the initial value is 1.

2. The graph represents Ava’s travel plans is graph (I).

The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. The net change in y coordinate is Δy, while the net change in the x coordinate is Δx. So the change in y coordinate with respect to the change in x coordinate can be written as,

m = Δy/Δx

where, m is the slope

Note that tan θ = Δy/Δx

We also refer this tan θ to be the slope of the line.

1. We have the coordinates as C(3, -2) and D(-2, 3).

So, the rate of change of linear function is

= 3 - (-2) / (-2 -3)

= 3+ 2 / (-5)

= 5/ (-5)

= -1.

and, the initial values is where the independent variable is zero which is (1, 0).

2. The graph represented for Ava journey is (A).

This, is because the speed of Ava car and speed of taxi is equal which is shown in graph 1 clearly.

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The given question is incomplete, complete question is:

1. A relation is plotted as a linear function on a coordinate plane starting at point C at (3, –2) and ending at point D at (–2, 3). What is the rate of change for the linear function and what is its initial value?

The rate of change is ______ and the initial value is ______.

A. 1 and -1

B. -1 and 1

C. 5 and -2

D. -2 and 5

2. Ava drove her car at a constant rate to the train station. At the train station, she waited for the train to arrive. After she boarded the train, she traveled at a constant rate, faster than she drove her car. She entered the taxi and traveled at a constant speed. This speed was equal to the speed at which she had driven her car earlier. After some time, she arrived at her destination.

Which graph represents Ava’s travel plans? (First 3 graphs are the options to this question.)

The problem statement is in Spanish and it asks to express the relationship between atoms and molecules for a metal sample containing [tex]4.25 moles[/tex] of molybdenum and [tex]1.63 moles[/tex] of titanium.

However, we can make some assumptions based on the typical behavior of metals. Metals usually exist in a solid state and consist of closely packed atoms arranged in a crystal lattice. Therefore, we can assume that the metal in question is solid, and its atoms are arranged in a regular pattern.

In this case, we can assume that the metal sample contains a mixture of molybdenum and titanium atoms, and the atoms are arranged in a crystal lattice structure. The ratio of moles of molybdenum to moles of titanium in the sample is approximately 2.61:1 (4.25/1.63), which means that there are more molybdenum atoms than titanium atoms in the sample.

Since the metal is solid, we can assume that the atoms are arranged in a crystal lattice, and the ratio of the number of atoms of each element in the crystal lattice is determined by the chemical formula of the compound. Without knowing the chemical formula, we cannot determine the exact ratio of atoms and molecules in the sample.

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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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The researchers would need to include at least 278 encounters in their sample to ensure that the standard deviation of the sample proportion is no larger than 0.03.

To determine the required sample size, we need to use the formula for the standard deviation of the sample proportion (σp):

[tex]\sigma_p = \sqrt{(p * (1 - p) / n)}[/tex]

where:

p is the estimated proportion (we don't have this information, so we'll use 0.5 as a conservative estimate for maximum variance),

n is the sample size.

Since the researchers want to ensure that the standard deviation of the sample proportion is no larger than 0.03, we can set up the following inequality:

0.03 ≥ √(0.5 * (1 - 0.5) / n)

Squaring both sides of the inequality to eliminate the square root:

0.03² ≥ 0.5 * (1 - 0.5) / n

0.0009 ≥ 0.25 / n

Now, solve for n:

n ≥ 0.25 / 0.0009

n ≥ 277.78

Since the sample size (n) must be a whole number, the researchers would need to include at least 278 encounters in their sample to ensure that the standard deviation of the sample proportion is no larger than 0.03. Rounding up, the required sample size is 278 encounters.

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The calculated value of the slope of the line is -7

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

The slope-intercept equation of a line is y = -7x - 2

This means that

y = -7x - 2

A linear equation is represented as

y = mx + c

Where

Slope = m

using the above as a guide, we have the following:

m = -7

This means that the slope of the line is -7

Hence, the slope of the line is -7

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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 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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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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The radius of the semi circle is 6 inches and the circumference of the semi circle is 18.84 inches.

A semi circle is half of a circle. The radius and circumference of the semi circle can be found as follows:

Therefore, radius is half of the diameter of a circle.

Hence,

radius of the semi circle  = 12 / 2

radius of the semi circle  =  6 inches

Therefore, let's find the circumference of the semi circle.

circumference of the semi circle = πr

circumference of the semi circle = 3.14 × 6

circumference of the semi circle =  18.84 inches

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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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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 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 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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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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Answer:

a. The surface area of the box needed to ship the containers with the least amount of cardboard is about 340.26 square inches.

b. The volume of the box needed to ship the containers is about 923.08 cubic inches.

(Hope this helps)

Step-by-step explanation:
a. To find the surface area of the box, we need to add up the areas of all the sides of the box. Think of wrapping the box in wrapping paper. The area of the paper that covers the box is the surface area.

b. To find the volume of the box, we need to measure how much space is inside the box. Think of the box as a swimming pool. The amount of water that can fit inside the pool is its volume.

The area of the region bounded by the spiral is 200π square units, and the length of the same spiral is 20π units.

a) To find the area of the region bounded by the spiral r = 20 for 0 ≤ θ ≤ π, we can use the polar coordinate system formula for area: Area = (1/2) ∫(r^2 dθ) from 0 to π

Given r = 20,
Area = (1/2) ∫((20)^2 dθ) from 0 to π
Area = (1/2) * 400 ∫(dθ) from 0 to π
Area = 200 [θ] from 0 to π
Area = 200(π - 0) = 200π square units.

b) To find the length of the same spiral (r = 20 for 0 ≤ θ ≤ π), we can use the formula for arc length in polar coordinates:
Arc Length = ∫(√(r^2 + (dr/dθ)^2) dθ) from 0 to π
Given r = 20 (a constant), dr/dθ = 0.
Arc Length = ∫(√((20)^2 + (0)^2) dθ) from 0 to π
Arc Length = 20 ∫(dθ) from 0 to π
Arc Length = 20[θ] from 0 to π
Arc Length = 20(π - 0) = 20π units.
So, the area of the region bounded by the spiral is 200π square units, and the length of the same spiral is 20π units.

Learn more about polar coordinate here: brainly.com/question/31422978

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