Question

The base of a solid is a circular disk with a radius of 8. Parallel cross-sections perpendicular to the base are squares. Find the volume of the solid.

a) 512 cubic units
b) 256 cubic units
c) 128 cubic units
d) 64 cubic units

Answer 1

a) The volume of the solid, a cylinder with square cross-sections perpendicular to its circular base, is 512 cubic units, obtained by integrating the areas of squares over its height. Thus the correct option is a.

Step-by-step explanation:

The solid in question is a cylinder with a circular base, and its cross-sections perpendicular to the base are squares. To find the volume of the solid, we need to consider the area of each square cross-section and integrate it over the entire height of the cylinder.

The area of a square is given by the formula
`\( \text{Area} = \text{side}^2 \)`. In this case, the side of the square is the diameter of the circular base, which is twice the radius (16 units).

Now, we integrate this area over the height of the cylinder. Since the height is not specified, let's denote it as h.

`\[ \int_(0)^(h) (16)^2 \, \text{d}h \]`

Evaluating this integral gives us
`\( (256)/(3)h^3 \).` Setting this equal to the volume of the cylinder
`(\( \pi r^2 h \))`, where r is the radius of the base, we get:

`\[ \pi \cdot 8^2 \cdot h = (256)/(3)h^3 \]`

Solving for h, we find
`\( h = 3 \)`. Substituting this back into the volume formula gives:

`\[ \text{Volume} = \pi \cdot 8^2 \cdot 3 = 512 \, \text{cubic units} \]`

Therefore, the correct volume of the solid is
`\( 512 \, \text{cubic units} \).`