A cylinder has a surface area of 2,512 square feet. The height of the cylinder is three times the radius of the base of the cylinder
Find the radius and the height of the cylinder
type the solution
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A cylinder has a surface area of 2,512 square feet. The height of the cylinder is three times the radius of the base of the cylinder
Find the radius and the height of the cylinder
type the solution
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Answer:
To find the radius and height of the cylinder, we can use the formula for the surface area, which is: A = 2πrh + 2πr^2. We know the surface area is 2,512 square feet, and that the height is three times the radius. Then we can substitute the values into the formula and solve for the radius and height:
2,512 sqft = 2πr(3r) + 2πr^2
Simplify: 7,540 = 6πr^2
Divide both sides by 6π: r^2 = 1
Let's find the radius and height of the cylinder.
We know that the surface area of a cylinder can be calculated using the formula:
Surface Area = 2πr² + 2πrh
Where:
- "r" is the radius of the base.
- "h" is the height of the cylinder.
In this problem, we're given that the height is three times the radius, so we can express the height as "3r."
The surface area of the cylinder is given as 2,512 square feet, so we have:
2,512 = 2πr² + 2πr(3r)
Now, we'll solve for "r" using this equation.
First, simplify the equation:
2,512 = 2πr² + 6πr²
Combine the terms with "r²":
2,512 = 8πr²
Now, isolate "r²" by dividing both sides by 8π:
r² = 2,512 / (8π)
Now, find "r" by taking the square root of both sides:
r = √(2,512 / (8π))
r ≈ 5 feet (rounded to the nearest foot)
Now that we have the radius, we can find the height using the relationship given:
h = 3r
h = 3 * 5
h = 15 feet
So, the radius of the cylinder is approximately 5 feet, and the height is 15 feet.
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