Question

calculator
what is the exact volume of the cylinder?
54π in³
108π in³
162π in³
324π in³
(there is a cylinder image with radius 6 in. and height 9 in.)

Explanation:

Step1: Recall cylinder volume formula

The formula for the volume \( V \) of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Step2: Identify radius and height

From the diagram, the radius \( r = 6 \) in and the height \( h = 9 \) in.

Step3: Substitute values into formula

Substitute \( r = 6 \) and \( h = 9 \) into the formula: \( V = \pi \times 6^2 \times 9 \).

Step4: Calculate \( 6^2 \)

\( 6^2 = 36 \).

Step5: Multiply to find volume

\( V = \pi \times 36 \times 9 = 324\pi \)? Wait, no, wait. Wait, 369 is 324? Wait, no, wait the options have 162π. Wait, maybe I misread the radius. Wait, the diagram: the dashed line is 6 in, is that the diameter? Wait, no, the dashed line is from center to edge, so radius is 6? Wait, no, maybe the 6 is the radius? Wait, no, let me check again. Wait, the cylinder: the top has a dashed line of 6 in, which is the radius? Wait, no, maybe the diameter? Wait, no, the height is 9 in. Wait, let's recalculate. If radius is 6, then \( r^2 = 36 \), height 9, so \( 36\times9 = 324 \), so \( V = 324\pi \)? But the options have 162π. Wait, maybe the 6 is the diameter? Wait, no, the dashed line is from center to edge, so radius. Wait, maybe I made a mistake. Wait, the options: 54π, 108π, 162π, 324π. Let's check again. Wait, maybe the radius is 3? No, the diagram shows 6 in as the radius. Wait, no, maybe the height is 3? No, the height is 9. Wait, no, let's recalculate. Wait, \( V = \pi r^2 h \). If r = 6, h = 9, then \( 6^2 = 36 \), 369 = 324, so \( V = 324\pi \)? But that's one of the options (324π in³). Wait, but maybe I misread the radius. Wait, maybe the 6 is the diameter? Then radius is 3. Then \( 3^2 = 9 \), 99 = 81, 81π, which is not an option. So maybe the radius is 6, height 9. Then 6²=36, 369=324, so 324π. But let's check the options again. The options are 54π, 108π, 162π, 324π. So 324π is an option. Wait, but maybe I made a mistake. Wait, no, let's do it again. Radius r = 6 in, height h = 9 in. Volume of cylinder is \( \pi r^2 h = \pi \times 6^2 \times 9 = \pi \times 36 \times 9 = 324\pi \) cubic inches. So the answer should be 324π in³. Wait, but let me check the diagram again. The cylinder has a radius of 6 in (dashed line from center to edge) and height 9 in. So yes, that's correct.

Answer:

324π in³ (the option with 324π in³)