Math  /  Geometry

QuestionFind the dimensions of the right circular cylinder described. The volume of a right circular cylinder is given by V=πr2hV=\pi r^{2} h.
The radius and height differ by two meters. The height is greater and the volume is 441π441 \pi cubic meters.
List the dimensions, separated by commas: \square

Studdy Solution

STEP 1

1. The cylinder is a right circular cylinder.
2. The volume of the cylinder is given by V=πr2h V = \pi r^2 h .
3. The height h h is greater than the radius r r by 2 meters.
4. The volume of the cylinder is 441π 441\pi cubic meters.

STEP 2

1. Express the height in terms of the radius.
2. Substitute the height expression into the volume formula.
3. Solve the resulting equation for the radius.
4. Calculate the height using the radius.
5. List the dimensions.

STEP 3

Express the height h h in terms of the radius r r :
h=r+2 h = r + 2

STEP 4

Substitute the expression for h h into the volume formula:
V=πr2h V = \pi r^2 h 441π=πr2(r+2) 441\pi = \pi r^2 (r + 2)

STEP 5

Solve the equation for r r :
441π=πr2(r+2) 441\pi = \pi r^2 (r + 2) 441=r2(r+2) 441 = r^2 (r + 2)
Expand and solve the cubic equation:
441=r3+2r2 441 = r^3 + 2r^2 r3+2r2441=0 r^3 + 2r^2 - 441 = 0
By trial or using a calculator, find r=7 r = 7 .

STEP 6

Calculate the height h h using the radius r=7 r = 7 :
h=r+2 h = r + 2 h=7+2=9 h = 7 + 2 = 9

STEP 7

List the dimensions of the cylinder:
Radius=7,Height=9 \text{Radius} = 7, \text{Height} = 9
The dimensions of the cylinder are:
7,9 \boxed{7, 9}

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