Algebra Question
Compute the Wronskian of the set {x, sin(2x), cos (2x)}.

Answer:

2 is really answer it is this question bro than ka for good point

Step-by-step explanation:

hello shreekant thanks 886A bubs 2 is answr

Using ratio, the number of boys in the class would be 230

If there are 5 boys for every 3 girls, then the ratio of boys to girls is 5:3. This means that for every 5 boys, there are 3 girls.

We can use this ratio to find the number of boys in a class of 368 students.

(5:3) × Total number of students

5/8 × 368 = 23

Therefore, there will be 230 boys in the class.

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The length of time that it will take for each of the workout sessions would be: Plan A = 1.5 hours, and Plan B =  0.5 hours.

The length of the sessions can be obtained by first obtaining drawing a system of equations for the two cases. On Wednesday, we can represent the sessions as follows:

2A + 12B = 9 hours

On Thursday, we would have

5A + 3B = 9 hours

2A + 12B = 9  Eqn 1

5A + 3B = 9   Eqn 2

Multiply equation 2 by -4

2A + 12B = 9

-20A - 12B = -36

2A - 20A = 9 -36

-18A = -27

Divide both sides by -18

A = 1.5 hours

For B, substitute A in the first equation

2(1.5) + 12B = 9

3 + 12B = 9

12B = 9 - 3

12B = 6

Divide both sides by 12

B = 0.5

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Complete Question:

Ann the trainer has two solo workout plans that she offers her clients: Plan A and Plan B. Each client does either one or the other (not both). On Wednesday there were 2 clients who did Plan A and 12 who did Plan B. On Thursday there were 5 clients who did Plan A and 3 who did Plan B. Ann trained her Wednesday clients for a total of 9 hours and her Thursday clients for a total of 9 hours. How long does each of the workout plans last?

Answer:

To determine which answer is bigger, let's calculate the values after the given changes and compare them.

Increasing 28 by 25%:

To calculate the increase, we need to find 25% of 28 and add it to 28.

25% of 28 = (25/100) * 28 = 7

Increased value = 28 + 7 = 35

Decreasing 40 by 15%:

To calculate the decrease, we need to find 15% of 40 and subtract it from 40.

15% of 40 = (15/100) * 40 = 6

Decreased value = 40 - 6 = 34

Comparing the results:

The increased value of 28 by 25% is 35, while the decreased value of 40 by 15% is 34.

Therefore, 35 is bigger than 34.

Answer: 8

Step-by-step explanation:

Rate of change is the slope

y = 8x + 15

follows format:

y = mx + b

m=8    so slope =8

The thermometer reading would be 73.48°C.

The boiling point of water is generally considered to be 100°C. According to the given information, the temperature of the liquid in the thermometer is 26.52°C lower than the boiling point of water. Therefore, to find the thermometer reading, we subtract 26.52 from 100.

100 - 26.52 = 73.48

Hence, the thermometer reading would be 73.48°C.

In this scenario, we are assuming that the thermometer is calibrated to measure temperature in Celsius. The boiling point of water at standard atmospheric pressure is 100°C, and the given information states that the liquid in the thermometer is 26.52°C lower than the boiling point.

By subtracting 26.52 from 100, we obtain a reading of 73.48°C.

Thermometers work by utilizing the principle that certain substances, such as mercury or alcohol, expand or contract with changes in temperature. The expansion or contraction is measured using a scale, which is marked with various temperature values.

In this case, the thermometer is calibrated in Celsius, so we refer to the Celsius scale. By subtracting 26.52 from 100, we find the temperature at which the liquid in the thermometer is settled, which is 73.48°C.

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The function  g(f(3)) = g(-31) = -65

Given that g(n) = 2n - 3 and f(n) = -n³ - 4. We need to find g(f(3)).Step-by-step explanation:

First, we need to find f(3),

we substitute n = 3 in f(n) = -n³ - 4

to getf(3) = -3³ - 4 = -27 - 4 = -31

Now, we substitute this value of f(3) in g(n) = 2n - 3

to getg(f(3)) = g(-31)We substitute n = -31 in g(n) = 2n - 3 to getg(-31) = 2(-31) - 3 = -62 - 3 = -65.

Note: The function f(n) = -n³ - 4 is a cubic polynomial, whereas g(n) = 2n - 3 is a linear polynomial.

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Answer:

  x = 6

Step-by-step explanation:

You want to find the value of x for similar triangles JKL and QRS, where JK=84, KL=91, QR=4x, RS=26.

Corresponding sides of similar triangles are proportional.

  JK/KL = QR/RS

  84/91 = 4x/26

  (26/4)(84/91) = x = 6

The value of x is 6.

__

Additional comment

The shorter:longer side length ratios in the two triangles are 12:13. This means 4x=24, or x=6.

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A normal distribution has a mean of 50 and a standard deviation of 3 , the probability P(X ≤ 44) = P(Z ≤ -2) = 0.0241  option a) 0.025.

In probability theory, normal distribution is also known as Gaussian distribution. It is a probability distribution that is symmetrical, bell-shaped, and a continuous probability distribution. It's also a part of continuous probability distribution that describes real-valued random variables whose probability density function is affected by two parameters: the mean μ and the variance σ².

Let us consider the problem. Suppose a normal distribution has a mean of 50 and a standard deviation of 3. Firstly, we need to standardize the random variable X that is to convert it to the standard normal distribution. We use the following formula for this Z = (X - μ) / σwhere X is the random variable and μ is the mean, σ is the standard deviation of the population.

So in this case, we can write this as Z = (44 - 50) / 3 = -2

We have now obtained the standard score or standard deviation for the random variable X.

Now we need to calculate the probability P(X ≤ 44) = P(Z ≤ -2).

The probability of Z being less than -2 is denoted by the area under the standard normal curve to the left of Z = -2.

Using the standard normal table we look for the probability that corresponds to -2 and the closest we find is 0.0228.

This probability represents the area under the standard normal distribution to the left of Z = -2.

To calculate the area to the left of Z = -2, we add the area to the left of the next integer, which is -3, which we find from the standard normal table as 0.0013, 0.0228 + 0.0013 = 0.0241.

Therefore, the probability P(X ≤ 44) = P(Z ≤ -2) = 0.0241 or 0.025 (rounded to three decimal places)Therefore, the answer is option A. 0.025.

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Answer:

12

Step-by-step explanation:

You simply add the coefficients together.
3 + 4 + 5 = 12

The equation of the perpendicular bisector of the line segment with endpoints (-7, 2) and (-1, -6) is y = (3/4)x + 1.

To find the equation of the perpendicular bisector of a line segment, we need to determine the midpoint of the line segment and the slope of the line segment. The perpendicular bisector will have a negative reciprocal slope compared to the line segment and will pass through the midpoint.

Given the endpoints (-7, 2) and (-1, -6), we can find the midpoint using the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Midpoint = ((-7 + (-1))/2, (2 + (-6))/2)

= (-8/2, -4/2)

= (-4, -2)

The midpoint of the line segment is (-4, -2).

Next, we need to find the slope of the line segment using the slope formula:

Slope = (y2 - y1)/(x2 - x1)

Slope = (-6 - 2)/(-1 - (-7))

= (-6 - 2)/(-1 + 7)

= (-8)/(6)

= -4/3

The slope of the line segment is -4/3.

Since the perpendicular bisector has a negative reciprocal slope, the slope of the perpendicular bisector will be 3/4.

Now, we can use the midpoint (-4, -2) and the slope 3/4 in the point-slope form of a line to find the equation of the perpendicular bisector:

y - y1 = m(x - x1)

y - (-2) = (3/4)(x - (-4))

y + 2 = (3/4)(x + 4)

y + 2 = (3/4)x + 3

y = (3/4)x + 1.

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The mean of a distribution is 217, while the median is 217.The statements is likely to be true about the distribution is  option  D. It is not symmetrical.

The distribution is likely to be asymmetrical if the mean and median have different values. If the mean and median of a distribution are equal, it is likely that the distribution is symmetrical, which means it has equal tail sizes on both sides. However, if the mean and median are not equal, it indicates that the distribution is skewed.

A negatively skewed distribution means that the mean is less than the median, whereas a positively skewed distribution means that the mean is greater than the median.In this case, since the mean and median of the distribution are equal, it is likely to be symmetrical. Since none of the options say "symmetrical," option D, "It is not symmetrical," is the correct answer.

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Answer:

x inches = 46 inches

2x inches = 92 inches

(7x - 4) inches = 318 inches

Step-by-step explanation:

The perimeter of the triangle is the sum of all three sides. Therefore,

x + 2x + (7x - 4) = 456 inches

10x - 4 = 456

10x = 460

x = 46 inches

Now, we find the other remaining sides.

2x inches = 2(46) inches = 92 inches

(7x - 4) inches = (7(46) - 4) inches = (322 - 4) inches = 318 inches

So, the final answer is

x inches = 46 inches

2x inches = 96 inches

(7x - 4) inches = 318 inches

The Transformation notation for transforming figure 1 to figure 2 is R y-axis T(-3, -2) R 180° T(2, -2).

The notation that describes the transformation of figure 1 to figure 2 is a combination of translation, reflection and rotation.To begin with, figure 2 is a mirror image of figure 1.

The mirror reflection transformation is represented by an ‘R’ in notation, followed by the line of reflection. In this case, the line of reflection is the y-axis. Therefore, the mirror reflection transformation notation is R y-axis.Next, the figure is translated or moved. The vertices (2, 3), (2, 6), (6, 6), (6, 4), (4, 4)

which forms the semi-cube have moved three units left and two units down. While the vertices (minus 2, minus 2), (minus 6, minus 2), (minus 6, minus 5), (minus 4, minus 4) and (minus 2, minus 4)

which forms the second semi-cube have moved two units right and two units down. Thus, the notation for this transformation is T(-3, -2) and T(2, -2).

Lastly, the figure is rotated 180 degrees counterclockwise around the origin. This transformation is represented by the notation R 180°.

Therefore, the transformation notation for transforming figure 1 to figure 2 is R y-axis T(-3, -2) R 180° T(2, -2).

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The value of the missing angle using Cosine rule is 128.68°

Let :

a = 7

b = 10

c = 4

Missing angle = B

From Cosine rule :

CosB = (a² + c² - b²)/2ac

Substituting the values into the formula :

CosB = (7² + 4² - 10²)/2×7×4

CosB = -35/56

CosB = -0.625

Taking the cosine inverse of 0.625

B = 128.68

Therefore, the value of the missing angle is 128.68

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This expression represents the Wronskian of the given set of functions {y1, y2}.Wronskian = e^2x * (cos(sqrt(x)) * sin(sqrt(x)) + (1/2)cos(sqrt(x)) * cos(sqrt(x)) - (1/2)sin(sqrt(x)) * sin(sqrt(x)))

To compute the Wronskian of a set of functions {y1, y2}, we need to evaluate the determinant of the matrix formed by the derivatives of these functions.

In this case, we have {y1, y2} = {e^x cos(sqrt(x)), e^x sin(sqrt(x))}.

To find the derivatives, we apply the chain rule. The derivatives of y1 and y2 with respect to x are:

y1' = (e^x cos(sqrt(x)))' = e^x cos(sqrt(x)) - (1/2)e^x sin(sqrt(x))

y2' = (e^x sin(sqrt(x)))' = e^x sin(sqrt(x)) + (1/2)e^x cos(sqrt(x))

Now, we form the matrix by arranging the derivatives:

| e^x cos(sqrt(x)) - (1/2)e^x sin(sqrt(x)) |

| e^x sin(sqrt(x)) + (1/2)e^x cos(sqrt(x)) |

Taking the determinant of this matrix, we have:

Wronskian = (e^x cos(sqrt(x)) - (1/2)e^x sin(sqrt(x))) * (e^x sin(sqrt(x)) + (1/2)e^x cos(sqrt(x)))

Expanding and simplifying the expression, we get:

Wronskian = e^2x * (cos(sqrt(x)) * sin(sqrt(x)) + (1/2)cos(sqrt(x)) * cos(sqrt(x)) - (1/2)sin(sqrt(x)) * sin(sqrt(x)))

Further simplification may not be possible without specific values of x, but this expression represents the Wronskian of the given set of functions {y1, y2}.

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Answer:

(a) 0
(b) 0
(c) 1
The function is not continous at x = 9

Step-by-step explanation:

To the left and right of the point x = 9, the graph would seem to continue at  f(x) = 0, however on the exact point x = 9 there is a hole in the graph allowing it to be equal to 1. Due to the hole in the graph, it is not continuous

Apologies if this is wrong its been a bit since I did calculus

Answer:

y = sec(x-2) - 1

Step-by-step explanation:

We can determine the graph is of a secant by the repeating parabolas. From there, graph sec(x) and add the transformations necessary to replicate the graph.

These are the two transactions that are given in the question and the journalizing of these transactions are explained with the appropriate steps and accounts.

The solution for the journalizing of the given transactions can be found below:

Journalizing refers to the recording of business transactions in the journal. In the process of journalizing, all transactions are recorded in the journal in a systematic way.

Journal is the first place where every transaction is recorded systematically.

The transaction is a business exchange in which one entity provides value to another entity in exchange for a specific item, service, or asset.

Transaction

(i) is paid Rs 2000 in cash as wages on the installation of the machine. This transaction can be journalized as follows: Journal of Transactions Date Particulars LF Amount (Dr.) Amount (Cr.) DD/MM/YYYCash a/c………………………Dr.2000To wages a/c2000 (Being cash paid as wages on the installation of the machine)Transaction

(ii) sold goods to Mr. Chopra at a list price of Rs 4,000. The sale is subject to a 10% trade discount and a 5% cash discount if the payment is made immediately, Mr. Chopra availed cash discount.

This transaction can be journalized as follows: Journal of Transactions Date Particulars LF Amount (Dr.) Amount(Cr.) DD/MM/YYYMr. Chopra a/c………………………Dr.3,420To Sales a/c4,000 (Being sales made to Mr. Chopra on a discount) To Trade Discount a/c400 (Being Trade Discount allowed to Mr. Chopra) Cash Discount a/c180 (Being cash discount allowed to Mr. Chopra) To Mr. Chopra a/c180 (Being cash received from Mr. Chopra) To CGST a/c90(Being CGST charged on sales) To SGST a/c90(Being SGST charged on sales)

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Answer:

6.6 inches

Step-by-step explanation:

Area of rectangle = L x W

A = 33 in²

W = 5 inches

L = ?

We Take

33 ÷ 5 = 6.6 inches

So, the length of the rectangle is 6.6 inches

To find the number of revolutions made by a roller,

We need to calculate the distance traveled by the roller in one revolution and then divide the total distance traveled by the roller by this distance.

The distance traveled in one revolution can be calculated using the formula:

Distance = Circumference of the roller = π * Diameter

Let's calculate it:

Diameter = 1.02 m

Radius = Diameter / 2 = 1.02 / 2 = 0.51 m

Circumference = 2 * π * Radius = 2 * 3.14159 * 0.51 ≈ 3.20876 m

Now, we divide the total distance traveled by the roller (291.72 m²) by the distance traveled in one (3.20876 m) to get the number of revolutions:

Number of Revolutions = Total Distance / Distance per Revolution

Number of Revolutions = 291.72 m² / 3.20876 m ≈ 90.972 revolutions

const diameter = 1.02; // meters

const thickness = 1.3; // meters

const area = 291.72; // square meters

const radius = diameter / 2;

const circumference = 2 * Math.PI * radius;

const totalDistance = area / thickness;

const revolutions = totalDistance / circumference;

console.log("Number of revolutions:", revolutions);

Therefore, the roller will make approximately 90.972 revolutions while rolling over the surface of 291.72 m².

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We find the number of revolutions a roller makes over a specific surface area by determining the area the roller covers in one revolution and dividing the total surface area by this figure. We treat the roller as a cylinder unrolling onto a plane and calculate the rectangle’s area that is formed.

To find the number of revolutions a roller makes when rolling over a specified surface area, we first need to find the area that the roller covers in one revolution. This can be obtained by imagining the roller as a cylinder 'unrolling' its outer curved surface onto the plane it's moving on. That unrolled surface forms a rectangle, whose area is equivalent to the circumference of the roller (which gives the length of the rectangle) multiplied by the thickness of the roller (which gives the width of the rectangle).

The circumference of the roller is given by the formula 2πr, where r is the radius, i.e., half of the diameter of 1.02m. That equates to 1.62m. The thickness of the roller, provided as 1.3m, serves as the width of the rectangle. So, the area covered in one revolution is 1.62m * 1.3m.

Dividing the total surface area covered, 291.72m², by the area covered in one revolution provides the number of revolutions the roller made.

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The angle z is in the third quadrant, the value of cosz is negative. Hence, cosz = -3/5.So, the value of cosz is -3/5.

Given sinz = -4/5 for pi < z < (3pi)/2, we need to find the value of cosz. We can use the trigonometric identity of Pythagorean theorem to find the value of cosz.

According to Pythagorean theorem, sin2θ + cos2θ = 1, where θ is the angle in the right-angled triangle and sin, cos are the trigonometric ratios.

The negative sign for the given sinz indicates that the angle z is in the third quadrant. So, we can take the help of the unit circle to find the value of cosz as shown below:

Here, we have used the Pythagorean identity of sin2z + cos2z = 1 on the unit circle to find the value of cosz. Since the value of sinz is already given, we can find the value of sin2z as: sin2z = sinz x sinz = (-4/5) x (-4/5) = 16/25

Then, we can substitute the value of sin2z in the Pythagorean identity as: cos2z = 1 - sin2z = 1 - (16/25) = 9/25We need to find the value of cosz.

So, we can take the square root of cos2z as: cosz = ±(√(9/25)) = ±(3/5)The sign of cosz can be determined by considering the quadrant of the angle z.

Since the angle z is in the third quadrant, the value of cosz is negative. Hence, cosz = -3/5.So, the value of cosz is -3/5.

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Answer: 12, 30, 162

Step-by-step explanation:

b)  For a regular polygon the central angles will all add up to 360

n= number of sides = 360/30

n=12

d) For exterior angles, the sum of the exterior angles is also 360

n = number of sides = 360/12

n=30

f) for any polygon

interior ange = ((n-2)180) / n

interior angle = ((20-2)180 / 20

interior angle =  (18)(180) / 20

interior angle = 162

Answer:

b. 12

d. 30

f. 162°

Step-by-step explanation:

b. The number of sides of a regular polygon can be found by dividing 360° by the central angle.

In this case, the central angle is 30°, so the number of sides is 360° / 30° = 12.

The polygon is a dodecagon.

d. The number of sides of a regular polygon can be found by dividing 360° by the exterior angle.

In this case, the exterior angle is 12°, so the number of sides is 360° / 12° = 30.

The polygon is a triacontagon.

f. The interior angle of a regular polygon can be found by using the formula  [tex]\frac{180(n-2) }{n}[/tex]where n is the number of sides.

In this case, n=20, so the interior angle is 180(20-2) / 20 = 162°.

Answer:

D: Resilient

Step-by-step explanation:

Below are the definitions for the words.

Durable- able to resist wear, decay, etc., well; lasting; enduring.

Colorful- abounding in color:

Flimsy- without material strength or solidity

Modern- of or relating to present and recent time; not ancient or remote

Resilient- returning to the original form or position after being bent, compressed, or stretched

From the definitions, we can see that durable most nearly means resilient.

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The probability that at least 2 of the colors will be the same, the following is the best simulation is Option D Put 5 pieces of paper in a bag, each labeled with a different color. For each trial, randomly select and replace a paper 7 times. Run 50 trials and find the fraction of times that at least two of the same color appear.

Given, Sienna buys a mystery pack of sparkly modeling clay. The pack has 5 colors, and there are 7 possible colors that are all equally likely. We have to find the probability that at least 2 of the colors will be the same. There are 5 colors in the pack of clay, and 7 colors are possible in total.

So, the probability of picking any color at random is

P(picking a color) = 5/7 = 0.7143Let P(X = 2) be the probability that exactly 2 of the colors are the same.

Then, the probability that at least 2 of the colors are the same can be calculated by using the formula:

P(X ≥ 2) = 1 - P(X < 2).

The simulation method that can be used to find the probability of at least 2 of the colors will be the same in the mystery pack is option D.

Put 5 pieces of paper in a bag, each labeled with a different color. For each trial, randomly select and replace a paper 7 times. Run 50 trials and find the fraction of times that at least two of the same color appear.

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Step 2:  He lines up his protractor with the baseline on the ray and the origin of the protractor on the vertex.

Based on the steps given, Jimmy accurately measures the angle by properly aligning the protractor's baseline with the ray.

Moreover, he positions the point of origin of the protractor at the vertex, which is the juncture where the two arms of the angle converge. By aligning the protractor correctly on the vertex, the measurement of the angle can be made more precise.

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The volume of the figure is 90 cubic meters

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

The figure

The volume of the figure is the product of the base area and the height

i.e. Volume = Base area * Height

Where, we have

Base area = 1/2 * 12 * 5

Base area = 30

So. we have

Volume = 30 * 3

Evaluate

Surface area = 90

Hence, the volume of the figure is 90 cubic meters

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a. The premium for this coverage will depend on the  individual's age, location, driving record, and the insurance company's rates.

b.   It will cost higher because the insurance company provides more coverage in the event of an accident.

c.  The difference in cost between the two coverage amounts depends on factors such as the insurance provider's rates and the person's specific circumstances.

a. Insurance coverage of 50/100/10:

The coverage limits 50/100/10 is a representation of  the minimum liability coverage required by state laws.

$50,000 = coverage for bodily injury liability per person.

$100,000  = coverage for bodily injury liability per accident.

$10,000 = coverage for property damage liability per accident.

b. Insurance coverage of 100/300/25:

The coverage limits 100/300/25 represent higher liability coverage limits:

$100,000  = coverage for bodily injury liability per person.

$300,000  = coverage for bodily injury liability per accident.

$25,000  = coverage for property damage liability per accident.

c. Difference in cost:

The difference in cost between the two coverage levels is influenced by a number of variables, including the insurance company's rates and the individual's unique situation.

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Answer:

∠ O = 55° , ∠ A = 90°

Step-by-step explanation:

2

the figure has opposite sides parallel and is therefore a parallelogram.

• opposite angles in a parallelogram are congruent, then

∠ O = ∠ Q = 55°

3

the sum of the interior angles of a quadrilateral is 360°

sum the angles and equate to 360°

∠ A + 70° + 110° + 90° = 360°

∠ A + 270° = 360° ( subtract 270° from both sides )

∠ A = 90°

Answer:

Step-by-step explanation:

b) Flip on its side so the triangle is the base

SA = Ph + 2B

P = perimeter of base

P = 8+6+hypotenuse

hypotenuse² =8²+6²

hypotenuse² = 100

hypotenuse = 10

P = 8+6+10

P= 24

h=height=5

B = area of base

B= 1/2 bh

B= 1/2 (8)(6)

B= 24

SA = Ph + 2B

SA = (24)(5) + 2(24)

SA = 168 yd²

d) SA (sphere) = 4[tex]\pi[/tex]r²

r=12

SA =  4[tex]\pi[/tex](12)²

SA = 576[tex]\pi[/tex] yd²

Answer:

b)  158 yd²

d)  576π yd²

Step-by-step explanation:

The surface area of a triangular prism is made up of 2 congruent triangles and 3 rectangles.

The area of a triangle is half the product of its base and height.

The area of a rectangle is the product of its width and length.

Therefore, to calculate the surface area of the given triangular prism, we first need to find the length of the hypotenuse (H) of the triangular base. To do this, we can use Pythagoras Theorem:

[tex]\begin{aligned}H^2&=6^2+8^2\\H^2&=36+84\\H^2&=100\\H&=\sqrt{100}\\H&=10\; \sf yd\end{aligned}[/tex]

Therefore, the surface area of the given triangular prism is:

[tex]\begin{aligned}\textsf{Surface Area}&=2 \cdot \left(\dfrac{1}{2} \cdot 6 \cdot 8\right)+(6 \cdot 5)+(8 \cdot 5)+(10 \cdot 5)\\\\&=2 \cdot \left(24)+30+40+50\\\\&=48+30+40+50\\\\&=158\; \sf yd^2\end{aligned}[/tex]

[tex]\hrulefill[/tex]

The formula for the surface area of a sphere is:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Surface area of a sphere}\\\\$SA=4 \pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

From the given diagram, the diameter of the sphere is 24 yd.

As the diameter is twice the radius, the radius of the sphere is r = 12.

Substitute the value of r into the formula to calculate the surface area of the sphere:

[tex]\begin{aligned}\textsf{Surface area}&=4 \pi (12)^2\\&=4 \pi (144)\\&=576\pi \; \sf yd^2\end{aligned}[/tex]

Therefore, the surface area of the sphere is 576π yd².