ANOVA F-statistic is defined as the Within Group Variation divided by the Between Group Variation. True False

Answer:

X= 95 Y=55

Step-by-step explanation:

95X5= 475

2X55= 110

475+110= 585

I hope this helps! : )

The probability that a random sample of 20 pies will have a mean baking time that exceeds 48 minutes is approximately 0.0037 or 0.37%

We can use the central limit theorem to approximate the distribution of the sample mean baking time. Since the sample size is large enough (20 pies) and the population standard deviation is known, we can use the normal distribution to approximate the distribution of the sample mean.

The mean of the sample mean baking time is the same as the population mean, which is 45 minutes.

To find the probability that a random sample of 20 pies will have a mean baking time that exceeds 48 minutes, we can standardize the sample mean using the formula:

[tex]z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]

= (48-45)/(5/√20)

= 2.68

Using a standard normal distribution table or calculator, we can find that the probability of a standard normal random variable being greater than 2.68 is approximately 0.0037. Therefore, the probability that a random sample of 20 pies will have a mean baking time that exceeds 48 minutes is approximately 0.0037 or 0.37%

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So, at 0 K, the value of the partition function is equal to the number of energy states plus one, which is 1.

The partition function (Z) is a concept used in statistical mechanics to describe the distribution of particles among different energy states. It is a sum of exponential terms, each corresponding to a different energy state. At absolute zero temperature (0 K), all particles are in their lowest energy state, and there is only one possible state with zero energy.

The partition function at 0 K is given by:

Z = Σ exp(-Ei/kT)

where Ei represents the energy of the ith state, k is the Boltzmann constant, and T is the temperature.

Since all particles are in the lowest energy state (E = 0), the term exp(-Ei/kT) becomes exp(0) = 1. Therefore, the partition function simplifies to:

Z = Σ (for i = 0 to N) 1 = N + 1

where N represents the total number of energy states.

At 0 Kelvin (absolute zero), the value of the partition function should be 1.

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The rejection region is t > 1.761.

To test the hypothesis that the mean fee for car washers (including tips) is greater than $12, we can perform a one-sample t-test.

Sample mean [tex]\bar{x}[/tex]  = $14.9

Sample standard deviation (s) = $6.75

Sample size (n) = 15

Significance level (α) = 0.05 (5%)

Since the sample size is small (n < 30) and the population standard deviation is unknown, we will use the t-distribution for inference.

Define the null and alternative hypotheses:

Null hypothesis (H₀): μ ≤ $12 (Mean fee for car washers is less than or equal to $12)

Alternative hypothesis (H₁): μ > $12 (Mean fee for car washers is greater than $12)

Determine the critical value (rejection region) based on the significance level and degrees of freedom.

The degrees of freedom (df) for a one-sample t-test is calculated as df = n - 1 = 15 - 1 = 14.

Using a t-table or statistical software, we find the critical t-value for a one-tailed test with α = 0.05 and df = 14 to be approximately 1.761.

Calculate the test statistic:

The test statistic for a one-sample t-test is given by:

t = ([tex]\bar{x}[/tex]  - μ) / (s / √n)

Plugging in the values:

t = ($14.9 - $12) / ($6.75 / √15) ≈ 2.034

Make a decision:

If the test statistic t is greater than the critical t-value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the calculated t-value (2.034) is greater than the critical t-value (1.761), indicating that it falls in the rejection region.

State the conclusion:

Based on the test results, at the 5% significance level, we have enough evidence to reject the null hypothesis.

We can infer that the mean fee for car washers (including tips) is greater than $12.

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A random variable is a numerical outcome that is generated by a probability experiment. It is a function that assigns a unique numerical value to each outcome of the experiment.

Random variables can be either discrete or continuous. Discrete random variables take on a countable number of distinct values, while continuous random variables can take on any value within a specified range. In statistical analysis, random variables are used to model the behavior of a system or population of interest. They are often used to describe the distribution of a population or the probability of different outcomes occurring in a given scenario. Random variables are an essential tool in probability theory, statistics, and other areas of mathematics and science.

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Therefore, The most general antiderivative of the function g(x) = cos(x) - 8sin(x) is F(x) = sin(x) + 8cos(x) + C. We can check the answer by differentiating F(x) which will give us g(x) = cos(x) - 8sin(x).

The most general antiderivative of the function g(x) = cos(x) - 8sin(x) is F(x) = sin(x) + 8cos(x) + C, where C is the constant of integration.
Explanation: To find the antiderivative of g(x), we use the formulae of integration of trigonometric functions. ∫cos(x) dx = sin(x) + C and ∫sin(x) dx = -cos(x) + C. Therefore, ∫cos(x) − 8sin(x) dx = ∫cos(x) dx − 8∫sin(x) dx = sin(x) + 8cos(x) + C. To check our answer, we differentiate F(x) with respect to x, we get g(x) = cos(x) - 8sin(x).

Therefore, The most general antiderivative of the function g(x) = cos(x) - 8sin(x) is F(x) = sin(x) + 8cos(x) + C. We can check the answer by differentiating F(x) which will give us g(x) = cos(x) - 8sin(x).

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It can be deduced as the final answer that about 45.23% of the time headways are less than 6 seconds and about 4.06% of the time headways are 10 seconds or greater.


Using the Poisson distribution with the mean rate λ, we can solve for the probability of no cars arriving in 20 seconds, which is:

P(X = 0) = e^(-λ) = 18/120

Solving for λ, we get:

λ = -ln(18/120) = 0.6052

Then we can use the Poisson distribution again to solve for the probability of exactly three cars arriving in 20 seconds, which is:

P(X = 3) = (λ^3 / 3!) * e^(-λ) ≈ 0.1097

Finally, we can multiply this probability by the total number of 20-second intervals to estimate the number of intervals in which exactly three cars arrive:

0.1097 * 120 ≈ 13.16

Therefore, we can approximate that 13 of the 120 intervals will have exactly three cars arrive.

The headway between vehicles is the time gap between the arrivals of two consecutive vehicles. We can estimate the percentage of time headways that are 10 seconds or greater and those that are less than 6 seconds by using the exponential distribution with the same mean rate λ as in problem 5.9.

For a headway X, the probability density function of the exponential distribution is given by:

f(x) = λ * e^(-λx)

Therefore, the probability of a headway being less than 6 seconds is:

P(X < 6) = ∫[0,6] λ * e^(-λx) dx = 1 - e^(-6λ)

Similarly, the probability of a headway being 10 seconds or greater is:

P(X ≥ 10) = ∫[10,∞) λ * e^(-λx) dx = e^(-10λ)

Using the value of λ obtained in problem 5.9, we can estimate these probabilities as:

P(X < 6) ≈ 0.4523 or 45.23%

P(X ≥ 10) ≈ 0.0406 or 4.06%

Therefore, we estimate that about 45.23% of the time headways are less than 6 seconds and about 4.06% of the time headways are 10 seconds or greater.

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The correlation coefficient between x and y is 0.61. The formula for the correlation coefficient (r) between two variables x and y is:

r = covariance(x, y) / (standard deviation(x) * standard deviation(y))

We are given that the covariance between x and y is 1225, the variance of x is 1600, and the variance of y is 2500. Since variance is the square of standard deviation, we can calculate the standard deviations of x and y as:

standard deviation(x) = sqrt(variance(x)) = √(1600) = 40

standard deviation(y) = sqrt(variance(y)) = √(2500) = 50

Plugging in these values into the formula for the correlation coefficient, we get:

r = 1225 / (40 * 50) = 0.61

r = 1225/(2000) = 0.61

Therefore, the correlation coefficient between x and y is 0.61.

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The minimum order of the Taylor polynomial centred at O for cos x required to approximate cos (-0.85) with an absolute error no greater than 10-4 is 5.

The Taylor series for cos x centred at O is given by:

cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ...

The nth-order Taylor polynomial for cos x centred at O is given by the first n terms of the Taylor series. We want to find the minimum n such that the absolute error between cos (-0.85) and the nth-order Taylor polynomial is no greater than 10-4.

The error term for the nth-order Taylor polynomial is given by:

Rn(x) = cos (c) * xn+1 / (n+1)!

where c is some value between 0 and x.

To find the minimum n, we need to find the value of n such that the error term is no greater than 10-4 for x = -0.85.

Substituting x = -0.85 into the error term and using the fact that |cos (c)| <= 1, we have:

|Rn(-0.85)| <= |(-0.85)^(n+1) / (n+1)!|

We want to find the minimum n such that the right-hand side is no greater than 10-4.

We can use a computer or calculator to find that n = 5 is the smallest integer that satisfies this condition. Therefore, the minimum order of the Taylor polynomial centred at O for cos x required to approximate cos (-0.85) with an absolute error no greater than 10-4 is 5.

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Answer:

write a story illustrate the saying that a lazy man goes to bed hungry

The data provided in the question represents annual sales of a product introduced 10 years ago. To analyze the data, the researcher first prepared a scatter plot to understand the relation between X and Y. It is observed that the relation is not completely linear but there is a positive correlation between X and Y.

To find an appropriate power transformation of Y, the Box-Cox procedure is used. The procedure evaluates SSE for different values of λ, including .3, .4, .5, .6, and .7. Standardization is applied to the data to obtain more accurate results. After evaluating SSE for all values of λ, it is suggested that the appropriate power transformation of Y is Y^0.5 (square root transformation).

Using the suggested transformation Y' = √Y, the researcher obtains the estimated linear regression function for the transformed data. The function is given by Y' = 7.58 + 0.221X.

In conclusion, the researcher used scatter plot analysis, Box-Cox procedure, and transformation techniques to analyze the annual sales data of the product introduced 10 years ago. The square root transformation of Y is found to be appropriate and the estimated linear regression function for the transformed data is obtained.

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The percentage of data that falls within three standard deviations of the mean is 99.7%.

The percentage of data that falls within three standard deviations of the mean is determined as follows;

one standard deviation below the mean = 34%

one standard deviation above the mean = 34%

So one standard deviation of the mean = 34% + 34% = 68%

Based on this information, we can conclude the following for a  normal distribution:

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The complete question is below:

For a normal distribution (shown in the diagram), what percentage of data falls within three standard deviations of the mean?

The first partial derivatives with respect to x, y, and z of the given function f(x, y, z) = 2x^2y − 9xyz/10yz^2 are:

fx = 4xy - (9yz/10z^2) = 4xy - (9/10z)

fy = 2x^2 - (9xz/10z^2) = 2x^2 - (9x/10z)

fz = (-9xy/5yz^2) - (18xyz/5yz^3) = (-9x/5z) - (18x/5y)

The partial derivative of a multivariable function with respect to a particular variable is calculated by considering all other variables as constants and differentiating with respect to the chosen variable. In this case, the partial derivative with respect to x involves differentiating the function with respect to x while treating y and z as constants, and similarly for y and z. The obtained partial derivatives are then used to find critical points, which are the points where all partial derivatives are zero.

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Answer:

16

Step-by-step explanation:

7 feet x 12 inches/foot = 84 inches

Then we divide the total length of the string (84 inches) by the length of each piece (5 inches) to get the total number of pieces:

84 inches / 5 inches per piece = 16.8 pieces

Since we cannot have a fractional number of pieces, we round down to the nearest whole number to get the final answer:

16 pieces

The equation of the tangent line:

y = (-x - sqrt(2)/2) - (π/2)(x - sqrt(2)/2)

To find the equation of the line tangent to the curve at the point corresponding to t = π/4, we need to find the first derivatives of x and y with respect to t, evaluate them at t = π/4, and then use these values to find the slope of the tangent line.

The first derivative of x with respect to t is:

dx/dt = -sint + tcost

The first derivative of y with respect to t is:

dy/dt = cost + tsint

Evaluating these at t = π/4, we get:

dx/dt|t=π/4 = -sqrt(2)/2

dy/dt|t=π/4 = (sqrt(2)/2) + (π/4)(sqrt(2)/2)

The slope of the tangent line is the ratio of the change in y to the change in x. So, the slope of the tangent line at t = π/4 is:

m = dy/dt|t=π/4 / dx/dt|t=π/4 = -1 - π/2

Now, we can use the point-slope form of the equation of a line to find the equation of the tangent line. Using the point (x(π/4), y(π/4)) = (sqrt(2)/2, sqrt(2)/2), we get:

y - (sqrt(2)/2) = (m)(x - sqrt(2)/2)

Substituting the value of m, we get:

y - (sqrt(2)/2) = (-1 - π/2)(x - sqrt(2)/2)

Expanding and simplifying, we get the equation of the tangent line:

y = (-x - sqrt(2)/2) - (π/2)(x - sqrt(2)/2)

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The algorithm that involves getting the next unsorted number and doing comparisons to determine its position within a sorted sublist is called the Insertion Sort algorithm.

The Insertion Sort algorithm works by maintaining a sorted sublist and repeatedly inserting the next unsorted element into the correct position within the sorted sublist.

Starting with the second element, it compares the element with the previous elements in the sorted sublist and shifts them to the right if they are greater, until finding the correct position to insert the element.

This process is repeated until all elements are sorted. Insertion Sort has a time complexity of O(n^2) in the worst case, making it efficient for small lists or partially sorted data.

Its main advantage is that it performs well for nearly sorted or small input sizes. The key idea of this algorithm is the comparison-based approach to find the correct position for each unsorted element within the sorted sublist.

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The experimental probability of rolling a 5 is 4/36, which simplifies to 1/9, or approximately 0.11.

We have,

The experimental probability of an event happening is the ratio of the number of times the event occurred to the total number of trials.

In this case,

The event is rolling a 5 and the total number of trials is the sum of the frequencies, which is 8 + 3 + 9 + 6 + 4 + 6 = 36.

The frequency of rolling a 5 is 4.

Therefore,

The experimental probability of rolling a 5 is 4/36, which simplifies to 1/9, or approximately 0.11.

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The function f(x) appears to be a polynomial function.

Based on the description of the graph, the function f(x) does not appear to be exponential, logarithmic, or rational.

Exponential functions typically exhibit a constant rate of change as x increases or decreases, resulting in a curve that either exponentially increases or decreases. The graph described does not match this pattern, as it increases in some areas and decreases in others.

Logarithmic functions have a characteristic shape with a vertical asymptote and a slow growth or decay. The given graph does not exhibit this behavior.

Rational functions are defined as the ratio of two polynomials, and their graphs often have vertical and horizontal asymptotes. However, the description does not mention any asymptotes, suggesting that the function is not rational.

The most suitable choice based on the given information is polynomial. Polynomial functions are characterized by having non-negative integer exponents and can exhibit various shapes, including increasing or decreasing trends. The description mentions that the graph is increasing in quadrant 3 and quadrant 4, indicating that the function could be a polynomial.

Without additional information or the specific equation of the function, it is challenging to determine the exact degree or form of the polynomial function.

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The simplified expression using a piecewise function:

f(x) = { x + 18, if x ≥ -18

{-x - 18, if x < -18

To simplify the expression |x - (-18)| without using the absolute value symbol, we need to consider two cases: when the expression inside the absolute value is positive (or equal to zero) and when it is negative.

When x - (-18) ≥ 0:

x - (-18) = x + 18, so in this case, the expression simplifies to x + 18.

When x - (-18) < 0:

Negate expression: -(x + 18) = -x - 18.

Now, we need to write the simplified expression using a piecewise function:

f(x) = { x + 18, if x ≥ -18

{-x - 18, if x < -18

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Thus, we can find nonzero matrices a, b, and c such that ac = bc and a is not equal to b by using the distributive property of matrix multiplication. One example of such matrices is provided above.

The problem statement requires us to find three matrices - a, b, and c, such that their product ac is equal to bc but a is not equal to b. To solve this problem, we need to use the properties of matrix multiplication. One such property is the distributive property, which states that a(b + c) = ab + ac, where a, b, and c are matrices.

Let's assume that a, b, and c are all 2x2 matrices. One example of such matrices could be:

a = [1 0]
   [0 2]

b = [2 0]
   [0 1]

c = [1 2]
   [3 4]

Using these matrices, we can verify that ac = bc, as follows:

ac = [1 0]  [1 2] = [1 2]
    [0 2]  [3 4]   [6 8]

bc = [2 0]  [1 2] = [2 4]
    [0 1]  [3 4]   [3 4]

As we can see, both products result in the same matrix. However, a and b are not equal, as a(1,1) = 1 and b(1,1) = 2. Therefore, we have found an example of three nonzero matrices such that ac = bc but a is not equal to b.

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Answer:  10

Step-by-step explanation: if you add all of them together you get 50 the oly one that can go into 50 without passing is 10

The expression that is equivalent to (2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3) is (D) (x - 1)/(x + 1)

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

(2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3)

When the expressions are factored, we have:

(2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3) = (2x - 3)(x + 5)/(3x + 1)(x + 5) * (3x + 1)(x - 1)/(x + 1)(2x - 3)

Cancelling out the common factors, we have

(2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3) = (x - 1)/(x + 1)

This means that the equivalent expression is (D) (x - 1)/(x + 1)

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The p-value for the given t-statistic falls between 0.005 and 0.01.

o find the p-value for the given t statistic of -2.63 with degrees of freedom of 24, we need to consult the t-distribution table or use statistical software.

Since the alternative hypothesis is μ < 100, we are conducting a one-tailed test in the left tail of the t-distribution. We want to find the area under the t-distribution curve to the left of -2.63.

Using the t-distribution table or software, we can determine that the p-value falls between 0.005 and 0.01. This means that the p-value for the statistic falls between 0.005 and 0.01.

Therefore, the p-value for the given t-statistic falls between 0.005 and 0.01.

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The statement is a suggestion or guideline for how to assign subjective probabilities.

Subjective probabilities are probabilities that are based on personal beliefs or judgments rather than on empirical evidence. Experience and intuition can be used to make informed guesses about the likelihood of an event occurring.

Available data can also be used to inform subjective probabilities, such as historical data or expert opinions. However, subjective probabilities may not always be reliable as they can be influenced by personal biases or limited information.

Therefore, it is important to use critical thinking and consider multiple sources of information when assigning subjective probabilities.

Additionally, subjective probabilities may be useful in situations where empirical data is not available or difficult to obtain, but they should be used with caution and always be subject to reassessment as new information becomes available.

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The length of the hypotenuse of this triangle is approximately 15.03 inches (rounded to the nearest tenth).

To find the length of the hypotenuse in a right triangle, we can use the trigonometric function cosine.

Given:

Leg length (adjacent side) = 7 in

Angle opposite the leg = 62°

We can use the cosine function, which relates the adjacent side and the hypotenuse of a right triangle:

cos(angle) = adjacent/hypotenuse

Let's substitute the known values into the equation:

cos(62°) = 7/hypotenuse

To solve for the hypotenuse, we rearrange the equation:

hypotenuse = 7/cos(62°)

Using a calculator, we find:

cos(62°) ≈ 0.4663

Now we can substitute this value into the equation:

hypotenuse = 7/0.4663

Calculating this, we get:

hypotenuse ≈ 15.03

Therefore, the length of the hypotenuse of this triangle is approximately 15.03 inches (rounded to the nearest tenth).

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P(X < 2) ≈ 0.1338 and P(X ≥ 2) ≈ 0.8662.

To find the probability P(X < 2), we need to calculate the probability of having fewer than 2 successes in 9 independent trials. This includes the cases where X is 0 or 1.

To find the probabilities P(X < 2) and P(X ≥ 2) in this scenario, we'll use the binomial distribution formula.

The binomial distribution formula is given by:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:

P(X = k) is the probability of getting exactly k successes,

n is the number of trials,

k is the number of successes,

p is the probability of success in a single trial, and

C(n, k) is the number of combinations of n items taken k at a time.

Given that n = 9 and p = 2/5, we can calculate the probabilities as follows:

P(X < 2):

P(X < 2) = P(X = 0) + P(X = 1)

P(X = 0):

P(X = 0) = C(9, 0) * (2/5)^0 * (1 - 2/5)^(9 - 0)

Using the combination formula C(9, 0) = 1:

P(X = 0) = 1 * 1 * (3/5)^9

P(X = 1):

P(X = 1) = C(9, 1) * (2/5)^1 * (1 - 2/5)^(9 - 1)

Using the combination formula C(9, 1) = 9:

P(X = 1) = 9 * (2/5) * (3/5)^8

Now we can calculate P(X < 2) by adding P(X = 0) and P(X = 1).

P(X ≥ 2):

P(X ≥ 2) = 1 - P(X < 2)

After calculating P(X < 2), we can subtract it from 1 to find P(X ≥ 2).

Let's perform the calculations:

P(X = 0) = 1 * 1 * (3/5)^9 = 0.01917808 (rounded to 8 decimal places)

P(X = 1) = 9 * (2/5) * (3/5)^8 = 0.11462304 (rounded to 8 decimal places)

P(X < 2) = 0.01917808 + 0.11462304 = 0.13380112 (rounded to 8 decimal places)

P(X ≥ 2) = 1 - P(X < 2) = 1 - 0.13380112 = 0.86619888 (rounded to 8 decimal places)

Therefore, P(X < 2) ≈ 0.1338 and P(X ≥ 2) ≈ 0.8662.

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If x = x1(t) and y = y1(t) and x = x2(t) and y = y2(t) are any two solutions of the linear nonhomogeneous system given by x' = p11(t)x + p12(t)y + g1(t), y' = p21(t)x + p22(t)y + g2(t), then x = x1(t) - x2(t) and y = y1(t) - y2(t) is a solution of the corresponding homogeneous system given by x' = p11(t)x + p12(t)y, y' = p21(t)x + p22(t)y.

To show that x = x1(t) - x2(t) and y = y1(t) - y2(t) is a solution of the corresponding homogeneous system, we need to verify that it satisfies the differential equations x' = p11(t)x + p12(t)y and y' = p21(t)x + p22(t)y with g1(t) = g2(t) = 0. Using the properties of derivatives, we can calculate that x' = x1'(t) - x2'(t) and y' = y1'(t) - y2'(t). Substituting these expressions and the expressions for x and y into the differential equations, we get:

x' = p11(t)x + p12(t)y

==> x1'(t) - x2'(t) = p11(t)(x1(t) - x2(t)) + p12(t)(y1(t) - y2(t))

==> p11(t)x1(t) + p12(t)y1(t) = p11(t)x2(t) + p12(t)y2(t)

y' = p21(t)x + p22(t)y

==> y1'(t) - y2'(t) = p21(t)(x1(t) - x2(t)) + p22(t)(y1(t) - y2(t))

==> p21(t)x1(t) + p22(t)y1(t) = p21(t)x2(t) + p22(t)y2(t)

Since x1(t), y1(t), x2(t), and y2(t) all satisfy the original nonhomogeneous system, we know that the expressions on the right-hand sides of the above equations are equal to g1(t) and g2(t), which are both zero in the corresponding homogeneous system. Therefore, x = x1(t) - x2(t) and y = y1(t) - y2(t) satisfy the differential equations x' = p11(t)x + p12(t)y and y' = p21(t)x + p22(t)y with g1(t) = g2(t) = 0, and hence they are a solution of the corresponding homogeneous system.

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The cooling rate of the turkey is 0.45°F per minute. This means that every minute, the temperature of the turkey decreases by 0.45°F.

The cooling rate of a roasted turkey can be determined by the rate at which it loses heat to its surroundings. In this case, the temperature of the turkey was 191°F when it was taken out of the oven and placed on a table in a room with a temperature of 75°F. After half an hour, the temperature of the turkey had decreased to 155°F.

To calculate the cooling rate, we can use Newton's law of cooling, which states that the rate of heat loss of an object is proportional to the difference in temperature between the object and its surroundings. The equation for Newton's law of cooling is:

dT/dt = -k (T - Ts)

where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the turkey at time t, Ts is the temperature of the surroundings (75°F), and k is a constant that depends on the specific heat of the turkey, its surface area, and other factors.

To solve for k, we can use the data given:

dT/dt = -k (T - Ts)

-36 = -k (155 - 75)

-36 = -k (80)

k = 0.45

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After considering all the data we conclude that the area of the region enclosed by the astroid is (3/8) π a⁴, under the condition that  x = a cos3θ, y = a sin3θ.

The astroid curve is given by x = a cos³θ, y = a sin³θ. The area enclosed by the astroid curve is given by the integral of ½ y dx from θ = 0 to θ = 2π ².

Staging x = a cos³θ and y = a sin³θ in ½ y dx, we get:

½ y dx = ½ a sin³θ (−3a sin²θ dθ) = −3/2 a⁴ cos⁶θ sin⁴θ dθ

Applying Integration to this expression from θ = 0 to θ = 2π provides us the area enclosed by the astroid curve:

A = ∫₀²π −3/2 a⁴ cos⁶θ sin⁴θ dθ

A = (3/8) π a⁴

Therefore, the area enclosed by the astroid curve is (3/8) π a⁴.
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