pls help with my homework

(1) The polynomial x^2 − 2 can be factored into (x + 5)(x − 5) in Z7[x].
(2) The polynomials x^3 − k, where k = 0, 1, 2, 3, 4, 5, 6, are all irreducible in Z7[x].

(3) The polynomial x^4 − 1 can be factored into (x − 1)(x + 1)(x^2 + 1) in Z7[x].

This is because 5 and −5 are both roots of the polynomial, since 5^2 ≡ 2 (mod 7) and (−5)^2 ≡ 2 (mod 7).

This is because none of them have any roots in Z7, which means they cannot be factored into lower degree polynomials.

For example, x^3 − 0 has no roots in Z7, since there is no integer x such that x^3 ≡ 0 (mod 7). Similarly, x^3 − 1 has no roots in Z7, since there is no integer x such that x^3 ≡ 1 (mod 7), and so on for the other values of k.

This is because 1, −1, and ±i are all roots of the polynomial, since 1^4 ≡ 1 (mod 7), (−1)^4 ≡ 1 (mod 7), and (±i)^4 ≡ 1 (mod 7). Therefore, x^4 − 1 = (x − 1)(x + 1)(x^2 + 1) in Z7[x].

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Given that it includes a quadratic term ([tex]b^{2}[/tex]), this formula reduces to 4b(b) = 0, which is a complex quantity.

A function with the formula a=0, b=1, and c=2 is known as a quadratic function. The function is known as the quadratic term (abbreviated as ax2), the linear term (abbreviated as bx), and the constant term (abbreviated as c).

The quadratic formula may be used to determine a parabola's axis of symmetry, the amount of real zeros in the quadratic equation, and the noughts of any parabola. It also produces the zeros of the any parabola.

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The statistical measures or five-number summary for the given data set are as follows:

In Mathematics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

In order to determine the five-number summary, we would arrange the data set in an ascending order:

4,6,6,8,11,12,13,17

Next, we would use an online graphing calculator to create the box plot as shown in the image attached below.

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The IikeIihood that the hawk wiII first take a songbird and subsequentIy a rabbit is therefore (2/9) x (8/11) = 16/99, or roughIy 0.1616.

In mathematics, a percentage is a number or ratio that may be stated as a fraction of 100. Divide a number by the whole and multiply the result by 100 to calculate its percentage. As a result, the percentage represents a part per hundred. 100% is meant by the phrase %. The symbol "%" is used to indicate it.

The probability that a hawk will eat a songbird before a rabbit is calculated as the product of those probabilities.

Given that there are four songbirds with eight rabbits, the IikeIihood that a songbird wiII be consumed by a hawk is 4/(4+6+8) = 4/18 = 2/9.

Three songbirds & eight rabbits remain after the hawk consumes a songbird, hence the IikeIihood that the hawk wiII eat a rabbit next is 8/(3+8) = 8/11.

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Answer:

Step-by-step explanation:

Well first we know that there are 25 squares. If we use some quick arithmetic, we can find how much each square represents out of 100%:

100/25 = 4

100% divided by 25 squares represents 4% for each square in the diagram. If we want to cover 48%, we can use some algebra:

4s = 48

s = 12

We need to shade in 12 squares to cover 48% of the diagram.

Hopefully this explanation was thorough enough!

Interval notation is (−∞, −4) U (−4, ∞)

The given inequality is ∣x + 4∣ > − 9. Step-by-step explanation: Absolute value inequalities: Inequalities that contain absolute values are known as absolute value inequalities. The absolute value inequality ∣x + 4∣ > − 9 is given.Inequality means that x can take any value except the value that makes the inequality false. If x satisfies the inequality, we write x ∈ (A, B) or x ∈ [A, B), where A and B are any two values that satisfy the inequality.The inequality ∣x + 4∣ > − 9 implies that the absolute value of x + 4 is greater than −9. The absolute value of x + 4 is always greater than or equal to zero.Therefore, the inequality can be written as∣x + 4∣ > 0This inequality implies that x is not equal to −4.The interval of x satisfying the given inequality is x ∈ (−∞, −4) U (−4, ∞), where U represents the union of two intervals. Therefore, the answer in interval notation is (−∞, −4) U (−4, ∞).Thus, the solution to the inequality |x + 4| > -9 in interval notation is (−∞, −4) U (−4, ∞).

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Answer: 24

Step-by-step explanation:

Answer:

60% of 44 is 26.4

Step-by-step explanation:

The solution to the equation (2)/(3)m=(5)/(8) is m = (15)/(16)

To solve the equation (2)/(3)m=(5)/(8), we can use the Division Property of Equality and the Multiplication Property of Equality.

Here are the steps:

1. Start with the original equation: (2)/(3)m=(5)/(8)

2. Multiply both sides of the equation by the reciprocal of (2)/(3), which is (3)/(2): (2)/(3)m * (3)/(2) = (5)/(8) * (3)/(2)

3. Simplify the left side of the equation: m = (5)/(8) * (3)/(2)

4. Simplify the right side of the equation: m = (15)/(16)

5. Check your solution by plugging it back into the original equation: (2)/(3) * (15)/(16) = (5)/(8)

6. Simplify the left side of the equation: (30)/(48) = (5)/(8)

7. Simplify the right side of the equation: (5)/(8) = (5)/(8)

8. Since both sides of the equation are equal, the solution is correct.

So the solution to the equation (2)/(3)m=(5)/(8) is m = (15)/(16).

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Answer:

3,200 Square centimeters

Step-by-step explanation:

In order to solve for the perimeter of a square with side lengths of 60, we can use this expression:

(We multiply by 4 because a square has four equal side lengths, and the perimeter is all of the side lengths added up)

Therefore, the perimeter of the square is 240cm.

Now that we know this, we can now take that perimeter, and subtract 160 from it.

(We subtract 160 from it because the length of the rectangle is 80cm, and because a rectangle has two sides that represent the length, we multiply the 80 by 2.)

Now, we can divide the 80 by 2 to get the length of the other 2 sides.

Therefore, the rectangles dimensions are 80cm by 40cm.

The area is the total space taken up by a flat (2-D) surface or shape. The area is always measured in square units.

To solve for the area of a rectangle, we use the expression:

Inserting the numbers into the equation:

Therefore, the area of the rectangle is 3,200 square centimeters.

Answer:

To simplify this expression, we need to first evaluate the terms inside the square root:

(-1)² = 1

(-2 - (-1))² = (-2 + 1)² = (-1)² = 1

Now we can substitute these values and simplify the expression:

√(6 - (-1)² + (-2 - (-1))²)

= √(6 - 1 + 1)

= √6

Therefore, the simplified form of the expression is √6.

Answer:

To evaluate the expression √(6-(-1)² + (-2-(-1)²), we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

1. First, we need to evaluate the expressions inside the parentheses:

-1² = (-1) × (-1) = 1

-2-(-1)² = -2-1 = -3

2. Next, we substitute these values into the expression:

√(6-(-1)² + (-2-(-1)²)

= √(6-1 + (-2-1))

= √(5 - 3)

= √2

Therefore, the value of the expression √(6-(-1)² + (-2-(-1)²) is √2.

Step-by-step explanation:

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We have the following response after answering the given question: As a equation result, Country A saw more erratic weather throughout the summer, with a 6°C difference in temperatures.

In a mathematical equation, the equals sign (=), which connects two claims and denotes equality, is utilised. In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a space. Mathematical expressions can be used to describe the relationship between the two sentences on either side of a letter. The logo and the particular piece of software frequently correspond. like, for instance, 2x - 4 = 2.

Which nation experienced the hottest summer?

We may examine the average temperature for each nation throughout the observed weekends to determine which nation experienced the hottest summer.

Country A: (21.4°C) (18+20+22+23+24)/5

(24+26+28+29+27)/5 = 26.8°C for Country B.

The summer was therefore hotter in Country B, with an average temperature of 26.8°C.

b) Over the summer, which nation saw more erratic weather?

We may examine the temperature ranges recorded for each nation to determine which experienced more erratic weather during the summer.

24°C - 18°C equals 6°C in Country A.

29°C - 24°C equals 5°C in country B.

As a result, Country A saw more erratic weather throughout the summer, with a 6°C difference in temperatures.

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The calculated probability that the selected member drinks champagne, given bridge is 82%

The statement derived from the question are stated as follows

The required conditional probability is calculated as

Probabiiity = Bridge and Drink/Bridge

So, we have the following equation

Probabiiity = 68%/83%

Evaluate

Probabiiity = 82%

Hence, the value of the probability is 82%

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The fund manager should invest $7.5 million in bonds and $27.5 million in mutual funds to maximize annual interest, and the maximum annual interest is given as  $2.595 million.

To solve this, we

Let x = amount invested in bonds and

y = amount invested in mutual funds.

We then can deduce the following constraints:

x + y ≤ 35 (total investment is at most $35 million)

x ≥ 5 (at least $5 million is invested in bonds)

y ≥ 20 (at least $20 million is invested in mutual funds)

100x + 200y ≤ 6000 (the total fees paid which cant exceed $6000)

Maximizing  the annual interest, which is given by:

0.06x + 0.09y

We will  use the method of Lagrange multipliers, to solve the optimization problems.

Let L(x, y, z) be the Lagrangian:

and we have that L(x, y, z) = 0.06x + 0.09y + z(x + y - 35) + μ(x - 5) + ν(y - 20) + ρ(100x + 200y - 6000)

We will have to take partial derivatives of L with respect to x, y, and z, and set them to be equals zero.

0.06 + z + μ + 100ρ = 0

0.09 + z + ν + 200ρ = 0

x + y - 35 = 0

Simplifying these equations, we have:

x = 7.5, y = 27.5, λ = -0.06, μ = 0, ν = 0, ρ = -0.02

0.06(7.5) + 0.09(27.5) = $2.595 million is the annual interest.

So then, It has been found that the fund manager should invest $7.5 million in bonds and $27.5 million in mutual funds to maximize annual interest, which is  $2.595 million.

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The scale factor that Georgia used her scale drawing is 3/8.

A scale factor is simply a ratio between two corresponding lengths, areas, or volumes of two similar geometric figures.

Given that, Georgia made a scale drawing of her apartment. Her bathroom is 3 inches in the drawing and 8 feet in real life.

To find the scale factor, we need to determine how many inches in the drawing represent 1 foot in real life. We can set up a proportion:

3 inches : 8 feet = x inches : 1 foot

Cross-multiplying, we get:

3 inches × 1 foot = 8 feet × x inches

Simplifying, we get:

3 = 8x

Dividing both sides by 8, we get:

x = 3/8

Therefore, the scale factor is 3/8, which means that for every 3 inches in the drawing, there is 8 feet in real life.

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The surface area of the cylinder with a radius of 1 ³/₄ and height of 3 ¹/₄ units will be 35.72 square units.

Let r be the radius and h be the height of the cylinder. Then the surface area of the cylinder will be given as,

SA = 2πrh square units

The radius is 1 ³/₄ units and the height is 3 ¹/₄ units.

First, convert the mixed fraction number into a fraction number. Then we have

1 ³/₄ = 7/4 units

3 ¹/₄ = 13/4 units

The surface area of the cylinder is calculated as,

SA = 2 x 3.14 x (7/4) x (13/4)

SA = 35.72 square units

The surface area of the cylinder with a radius of 1 ³/₄ and height of 3 ¹/₄ units will be 35.72 square units.

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The complete question is given below.

Exact surface area. The radius is 1 3/4 and the height is 3 1/4 of the cylinder.

Check the picture below.

so the base of the pyramid is a triangle whose base is 12 and altitude is "x", and the pyramid has a height/altitude of 15, so

[tex]\textit{volume of a pyramid}\\\\ V=\cfrac{1}{3}Bh ~~ \begin{cases} B=\stackrel{base's}{area}\\ h=height\\[-0.5em] \hrulefill\\ B=\frac{1}{2}(12)(x)\\[1em] h=15\\ V=240 \end{cases}\implies 240=\cfrac{1}{3}\left[\cfrac{1}{2}(12)(x) \right](15) \\\\\\ 240=30x\implies \cfrac{240}{30}=x\implies 8=x[/tex]

I - The collapse of the housing market due to low interest rates, easy credit, insufficient regulation, and toxic subprime mortgages: C. Problems in financial markets.

II - Consumers are very worried about the economy: B. Negative demand shift.

1 - The collapse of the housing market due to low interest rates, easy credit, insufficient regulation, and toxic subprime mortgages falls into the category of "Problems in financial markets" (C). This event can lead to a recession because it can cause a decrease in the wealth of households, leading to a decrease in consumer spending and a decrease in aggregate demand.

II - Consumers being very worried about the economy falls into the category of "Negative demand shift" (B). This event can lead to a recession because when consumers are worried about the economy, they tend to save more and spend less, leading to a decrease in aggregate demand. This decrease in demand can cause businesses to decrease production and lay off workers, leading to a decrease in income and further decrease in demand.

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If the mean of five scores is 27, then the number 33 must be added to the previous five scores to make the mean of six scores as 28.

It is given that mean of five scores is 27. If the unknown number x is added to the five scores, the new mean becomes 28. To calculate the value of x, the following equations are considered.

Let us say that the five scores are A, B, C, D and E.

∴ Mean of five scores = (A+B+C+D+E)/ 5 = 27

⇒ (A+B+C+D+E) = 27 × 5 = 135

Now, if sixth score x is added, then mean score becomes 28.

⇒ (A+B+C+D+E+x)/ 6 = 28

⇒ (A+B+C+D+E+x) = 28×6 = 168

Putting the value of (A+B+C+D+E) in above equation, we get:

135 + x = 168

x = 168 - 135 = 33

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The median is 138.

What is the median?

The median is the value that divides a data sample, a population, or a probability distribution's upper and lower halves in statistics and probability theory. It could be referred to as "the middle" value for a data set.

Here, we have

Given: data set:

135, 149, 156, 112, 134, 141, 154, 116, 134, 156.

To get the box plot we begin by arranging the data in ascending order:

135, 149, 156, 112, 134, 141, 154, 116, 134, 156

rearranging the data set we get:

112,116, 134, 134, 135, 141, 149, 154, 156, 156

then:

Lower value = 112

Q1 = 134

Median = (135+141)/2 = 138

Q3 = 154

Largest value =156

Hence, the median is 138.

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In response to supplied query, we may state that The connection shown in the table is represented by a linear equation in the slope-intercept form.

In mathematics, the intersection point is where the line's slope intersects the y-axis. a point on a line or curve where the y-axis crosses. The equation for the straight line is given as Y = mx+c, where m denotes the slope and c the y-intercept. In the intercept form of the equation, the line's slope (m) and y-intercept (b) are highlighted. The slope and y-intercept of an equation with the intercept form (y=mx+b) are m and b, respectively. There are several equations that may be rewritten to seem to be slope intercepts. The slope and y-intercept are both modified to 1 if y=x is rewritten as y=1x+0, for example.

We must know the slope and the y-intercept of the line in order to find the slope-intercept form of a linear equation.

The first and last points in the table should be chosen:

slope = [tex](5 - (-5)) / (3 - (-2)) = 10/5 = 2[/tex]

Now that we have the slope, we can write the equation of the line using the point-slope form of a linear equation:

y - y1 = m(x - x1) (x - x1)

Each position along the line may be chosen as (x1, y1). Let's pick point number one (3, -5):

[tex]y - (-5) = 2(x - 3) (x - 3)\\y + 5 = 2x - 6\\y = 2x - 11[/tex]

The connection shown in the table is represented by a linear equation in the slope-intercept form.

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A basis for the subspace  of R4 defined by the equation−2x1​−7x2​+2x3​−4x4​=0 is the set of vectors {(-7/2, 1, 0, 0), (1, 0, 1, 0), (-2, 0, 0, 1)}.

A basis for the subspace of −2x1​−7x2​+2x3​−4x4​=0 can be found by solving for one of the variables in terms of the others and then finding the general solution.

First, solve for x1 in terms of the other variables:
-2x1 = 7x2 - 2x3 + 4x4
x1 = (-7/2)x2 + x3 - 2x4

Now, the general solution can be written as:
(x1, x2, x3, x4) = (-7/2)x2 + x3 - 2x4, x2, x3, x4

This can be rewritten in terms of the free variables x2, x3, and x4:
(x1, x2, x3, x4) = x2(-7/2, 1, 0, 0) + x3(1, 0, 1, 0) + x4(-2, 0, 0, 1)

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Answer:

2.4.1. To calculate how many litres of fuel are needed to travel 400km, we need to use the given fuel consumption rate of 8L/100km:

Fuel consumed for 400km = (8/100) x 400 = 32 litres

Therefore, 32 litres of fuel are needed to travel 400km.

2.4.2. To calculate how far you can travel with 50L of fuel, we can rearrange the formula from part 2.4.1 to solve for distance:

distance = (fuel consumed / fuel consumption rate) x 100

Plugging in the values we have:

distance = (50 / 8) x 100 = 625 km

Therefore, with 50 litres of fuel, you can travel 625 km.

Answer:

3^2 = 9

So 9 is the number.

Boulder takes 4.13 seconds to reach its maximum height.

The boulder's maximum height is 282.38 feet.

Velocity is a physical quantity that describes the rate of change of an object's position over time. Velocity has both magnitude as well as direction that's why it is a vector quantity.

Given that:

Upward velocity of the boulder = 132 ft/s

Function for the height of the boulder as a function of time, t:

h(t) = -16[tex]t^{2}[/tex] + 132t + 30

We can find the maximum height of the boulder and the time it takes to reach that height by finding the vertex of the parabolic function h(t) using the formula:

t = -b/2a

where a = -16 and b = 132.

First, we need to find the time it takes the boulder to reach its maximum height, t:

t = -b/2a = -132/(2*(-16)) = 4.125 seconds

So, it takes the boulder 4.125 seconds to reach its maximum height.

Next, we need to find the maximum height of the boulder, h:

h = h(t) = -16t^2 + 132t + 30 = -16(4.125)^2 + 132(4.125) + 30 = 282.375 feet

So, the boulder's maximum height is 282.375 feet.

Rounding to the nearest hundredth, the time it takes the boulder to reach its maximum height is 4.13 seconds and the boulder's maximum height is 282.38 feet.

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Answer:

To find 3.5% of 21, we can convert 3.5% to a decimal by dividing by 100:

3.5% = 3.5/100 = 0.035

Then, we can multiply 0.035 by 21 to find the answer:

0.035 * 21 = 0.735

Therefore, 3.5% of 21 is 0.735.

0.735

because o.31 percent of 21 is 0.735

Answer:

245

Step-by-step explanation:

There are 6 faces in a cube, the total surface area of the toy box is 6 x 4 = 24 square feet. So correct option is B.

In geometry, a cube is a three-dimensional shape that is bounded by six square faces, with each face meeting at right angles. A cube is a regular polyhedron, which means that all of its faces are congruent and all of its edges have the same length.

The cube has a total of eight vertices (corners) and twelve edges, with each vertex connecting three edges and each edge connecting two vertices. The volume of a cube can be calculated using the formula V = s^3, where s is the length of one edge.

Cubes are commonly used in mathematics, physics, and engineering to represent three-dimensional objects and to model various phenomena. For example, cubes can be used to represent the atoms in a crystal lattice, the cells in a grid, or the pixels in a digital image

The toy box is a cube, and each edge measures 2 feet. So, the surface area of one face of the cube is 2 x 2 = 4 square feet. Since there are 6 faces in a cube, the total surface area of the toy box is 6 x 4 = 24 square feet.

Therefore, the answer is (B) 24 feet squared.

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The complete question is:

If three pens are randomly selected from the following basket of pens, without replacement,  the probability that all three pens are blue is 0.027, none of the pens are blue is 0.343, and  at least one of the pens is black is 0.657.

The probability of selecting three blue pens without replacement from the basket of pens is:  

P(all 3 blue) = (Number of blue pens / Total number of pens)³

= (3 / 10)³

= 0.027

The probability of selecting three pens that are not blue without replacement from the basket of pens is:  

P(none blue) = (Number of pens that are not blue / Total number of pens)3

= (7 / 10)³

= 0.343

The probability of selecting at least one black pen without replacement from the basket of pens is:

P(at least one black) = 1 - (Number of pens that are not black / Total number of pens)3

= 1 - (7 / 10)³

= 0.657

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The polynomial as a product of linear factor [tex]9(x) = x4 - 2x3 + 5x2 8x + 4 g(x)[/tex] are g(x) = (x-2)(x-1)(x+1)(x+4) , all the zeros of function are 2,1,-1,-4.

In order to write the polynomial as a product of linear factors, we must first find its zeros. The zeros of a polynomial are the values of x that make the polynomial equal to zero. The way to find the zeros is to set the polynomial equal to zero, and solve for x.

For this particular polynomial, the equation would be: x^4 - 2x^3 + 5x^2 + 8x + 4 = 0.

We can solve this equation by factoring. When factoring, we look for common factors among the terms and group them together. After factoring, the equation becomes: (x - 2)(x - 1)(x + 1)(x + 4) = 0.

The zeros of the equation are x = 2, 1, -1, -4. This means that the polynomial can be written as the product of linear factors, which is (x-2)(x-1)(x+1)(x+4). The zeros of this function are x = 2, 1, -1, -4.

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The polynomial as the product of linear factors. 9(x) = x4 - 2x3 + 5x2 8x + 4 g(x) is a product of (x - 2)(x - 1)(x + 1)(x + 4) . All the zeros of function are x = 2,1,-1,-4.

In order to write the polynomial as a product of linear factors, we must first find its zeros. The zeros of a polynomial are the values of x that make the polynomial equal to zero. The way to find the zeros is to set the polynomial equal to zero, and solve for x.

For this particular polynomial, the equation would be: x^4 - 2x^3 + 5x^2 + 8x + 4 = 0.

We can solve this equation by factoring. When factoring, we look for common factors among the terms and group them together. After factoring, the equation becomes: (x - 2)(x - 1)(x + 1)(x + 4) = 0.

The zeros of the equation are x = 2, 1, -1, -4. This means that the polynomial can be written as the product of linear factors, which is (x-2)(x-1)(x+1)(x+4).

The zeros of this function are x = 2, 1, -1, -4.

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The lateral and surface area of the given shape above in terms of π would be = 48π yd²:120π yd². That is option A.

To calculate the lateral surface area of the cylinder the formula below is used:

Lateral surface area = 2πrh

where;

r = diameter/2 = 12/2 = 6 yd

h = 4 yd

Lateral surface area = 2 ×π × 6×4 = 48π yd²

To calculate the surface area of the cylinder the following formula is used:

Surface area = 2πrh + 2πr²

r = diameter/2 = 12/2 = 6 yd

h = 4 yd

surface area = (2×π×6×4)+(2×π×36)

= 48π+72π

= 120π yd²

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