Question

height of 6 units, what is the volume of the empty space between the cylinder and the box? round the answer to two decimal places
a. 20.60 cubic units
b. 37.77 cubic units
c. 51.27 cubic units
d. 75.40 cubic units

Explanation:

Step1: Determine the shape of the box (cube)

Assume the box is a cube (since the cylinder is inscribed, the side length of the cube equals the diameter of the cylinder's base and the height of the cube equals the height of the cylinder). Let's assume the diameter of the cylinder's base (and side length of the cube) is \( d \), and height \( h = 6 \) units. Wait, maybe the radius is \( r \), but actually, from the diagram, the box is a square prism (cube-like) with side length equal to the diameter of the cylinder. Wait, maybe the cylinder has a diameter equal to the side length of the box, so let's say the side length of the box (let's call it \( s \)) is equal to the diameter of the cylinder, so radius \( r=\frac{s}{2} \). But maybe the box is a cube with side length equal to the height? Wait, no, the height of the box is 6 units, so the height of the cylinder is also 6 units. Let's assume the base of the box is a square with side length equal to the diameter of the cylinder. Wait, maybe the cylinder has a radius \( r \), so diameter \( 2r \), and the box has length, width \( 2r \), and height 6. Then volume of the box \( V_{box}= (2r) \times (2r) \times 6 = 24r^2 \). Volume of the cylinder \( V_{cylinder}= \pi r^2 h = \pi r^2 \times 6 = 6\pi r^2 \). Then empty space \( V_{empty}= V_{box}- V_{cylinder}=24r^2 - 6\pi r^2 = 6r^2(4 - \pi) \). Wait, but we need to find \( r \). Wait, maybe the side length of the box is equal to the height? No, the height is 6. Wait, maybe the box is a cube with side length 6? Wait, if the box is a cube with side length 6, then the cylinder inscribed in it has diameter 6, so radius 3. Then:

Step2: Calculate volume of the box (cube)

Volume of cube \( V_{box}= s^3 \), where \( s = 6 \). So \( V_{box}= 6 \times 6 \times 6 = 216 \)? Wait, no, that can't be, because the options are much smaller. Wait, maybe I made a mistake. Wait, the height is 6 units, but maybe the base of the box is a square with side length equal to the diameter of the cylinder, and the height is 6. Wait, maybe the cylinder has a radius of 2? No, let's re-examine. Wait, the options are around 20 - 75, so the volume of the box must be smaller. Wait, maybe the side length of the box is 6? No, that would make the box volume 216, cylinder volume \( \pi r^2 h \), if radius is 3 (diameter 6), then cylinder volume is \( \pi \times 3^2 \times 6 = 54\pi \approx 169.65 \), then empty space would be \( 216 - 169.65 = 46.35 \), which is not among the options. Wait, maybe the box is not a cube, but a rectangular prism with length, width equal to the diameter of the cylinder, and height 6. Wait, maybe the diameter of the cylinder is equal to the height? No, the height is 6. Wait, maybe the radius is 2? No, let's check the options. Let's suppose the side length of the box is 4? No, the options are 20.60, 37.77, 51.27, 75.40. Wait, maybe the box is a square prism with length and width equal to the height (6), so length=width=6, height=6? No, that's a cube. Wait, no, maybe the cylinder has a diameter equal to the height, so diameter 6, radius 3, but then box volume is 6*6*6=216, cylinder volume 54π≈169.65, empty space≈46.35, not matching. Wait, maybe the side length of the box is 4? No, let's think again. Wait, maybe the box is a square prism with length and width equal to the diameter of the cylinder, and height 6, but the diameter is equal to the height? No, height is 6, so diameter 6, radius 3. Wait, maybe I messed up the shape. Wait, the box is a rectangular prism (a box) with length, width, and height. The cylinder is inside, so the length and width of the box are equal to the diameter of the cylinder's base, and the height of the box is equal to the height of the cylinder. Let's assume the diameter of the cylinder's base is \( d \), so radius \( r = d/2 \), height \( h = 6 \). Then volume of the box: \( V_{box}= d \times d \times 6 = 6d^2 \). Volume of the cylinder: \( V_{cylinder}= \pi r^2 h = \pi (d/2)^2 \times 6 = \pi d^2 \times 6 /4 = (3\pi d^2)/2 \). Then empty space: \( V_{empty}= 6d^2 - (3\pi d^2)/2 = d^2(6 - (3\pi)/2) = (3d^2/2)(4 - \pi) \). Now, we need to find \( d \). Wait, maybe the height of the box is equal to the diameter? No, the height is 6. Wait, maybe the box is a cube with side length equal to the height, so \( d = 6 \). Then:

\( V_{box}= 6 \times 6 \times 6 = 216 \) (no, that's too big). Wait, the options are small, so maybe the side length of the box is 4? No, let's check the options. Wait, maybe the cylinder has a radius of 2, so diameter 4, box length and width 4, height 6. Then:

Volume of box: \( 4 \times 4 \times 6 = 96 \)

Volume of cylinder: \( \pi \times 2^2 \times 6 = 24\pi \approx 75.40 \)

Empty space: \( 96 - 75.40 = 20.60 \). Wait, that's option A. But wait, 24π is about 75.40, 96 - 75.40 = 20.60. But let's check with radius 3:

Box volume: 6*6*6=216, cylinder volume: π*3²*6=54π≈169.65, empty space≈46.35 (not an option). Radius 2: box volume 4*4*6=96, cylinder 24π≈75.40, empty 20.60 (option A). But wait, maybe the box is a square with side length equal to the height, so height 6, length and width 6? No, that's a cube. Wait, maybe the diagram shows the box as a square prism with side length equal to the diameter of the cylinder, and height 6. If the diameter is 4 (radius 2), then:

Box volume: 4*4*6=96

Cylinder volume: π*2²*6=24π≈75.40

Empty space: 96 - 75.40=20.60 (option A). But wait, another approach: maybe the box is a cube with side length equal to the height, 6, but the cylinder has diameter equal to the side length, so radius 3. Then box volume 6³=216, cylinder volume π*3²*6=54π≈169.65, empty space 216-169.65=46.35 (not an option). So maybe the side length of the box is 4, height 6. Then as above, empty space 20.60 (option A). Wait, but let's check the options. Option A is 20.60, B 37.77, C 51.27, D 75.40. Wait, maybe I made a mistake in the box shape. Wait, the box is a rectangular prism with length, width, and height, and the cylinder is inside, so the length and width are equal to the diameter of the cylinder, and height equal to the cylinder's height. Let's suppose the diameter of the cylinder is 4 (so radius 2), height 6. Then:

Box volume: 4*4*6=96

Cylinder volume: π*2²*6=24π≈75.40

Empty space: 96 - 75.40=20.60 (option A). But wait, another possibility: maybe the box is a square with side length equal to the radius? No, that doesn't make sense. Wait, maybe the height of the box is equal to the radius? No, the height is 6. Wait, maybe the cylinder has a radius of 1, diameter 2, box volume 2*2*6=24, cylinder volume π*1²*6=6π≈18.85, empty space 24-18.85=5.15 (not an option). So the first calculation with radius 2, diameter 4, box volume 96, cylinder 75.40, empty 20.60 (option A) matches. Wait, but let's check the formula again. Wait, maybe the box is a cube with side length 6, but the cylinder has a diameter of 6, so radius 3. Then box volume 6*6*6=216, cylinder volume π*3²*6=54π≈169.65, empty space 216-169.65=46.35 (not an option). So the correct assumption must be that the side length of the box (length and width) is equal to the diameter of the cylinder, and the height is 6, and the diameter is 4 (so radius 2). Then empty space is 20.60, which is option A. Wait, but let's check the answer options. Option A is 20.60, which matches 96 - 24π ≈96 -75.40=20.60. So:

Step1: Volume of the box (rectangular prism)

Assume the box has length = width = 4 (diameter of cylinder), height = 6. Then \( V_{box}= 4 \times 4 \times 6 = 96 \).

Step2: Volume of the cylinder

Cylinder has radius \( r = 2 \), height \( h = 6 \). \( V_{cylinder}= \pi r^2 h = \pi \times 2^2 \times 6 = 24\pi \approx 75.40 \).

Step3: Volume of empty space

\( V_{empty}= V_{box}- V_{cylinder}= 96 - 75.40 = 20.60 \).

Answer:

A. 20.60 cubic units