Question

Which of the following has the smallest volume?

(A) A cone with a radius of 4.5 units and a height of 2 units
(B) A cylinder with a radius of 3 units and a height of 2.5 units
(C) A hemisphere with a radius of 4 units
(D) A sphere with a radius of 2.5 units

Answer 1

By calculating the volumes of a cone, cylinder, hemisphere, and sphere using their respective formulas, we find that the sphere with radius 2.5 units has the smallest volume of all the given options.

Step-by-step explanation:

To compare the volumes of the given shapes, we will use the formulas for the volume of each shape. For the cone, cylinder, hemisphere, and sphere, we use the formulas
`V = \( (1)/(3) \pi r^2 h \)` for a cone,
`V = \( \pi r^2 h \)` for a cylinder,
`V = \( (2)/(3) \pi r^3 \)` for a hemisphere, and
`V = \( (4)/(3) \pi r^3 \)` for a sphere.

Now let's calculate the volume for each option:

• A cone with radius 4.5 units and height 2 units:
  `\( V = (1)/(3) \pi * (4.5)^2 * 2 \) units 3`
• A cylinder with radius 3 units and height 2.5 units:
  `\( V = \pi * (3)^2 * 2.5 \) units 3`
• A hemisphere with radius 4 units:
  `\( V = (2)/(3) \pi * (4)^3 \) units 3`
• A sphere with radius 2.5 units:
  `\( V = (4)/(3) \pi * (2.5)^3 \) units 3`

Let's compute these volumes:

• For the cone:
  `\( V = (1)/(3) \pi * 20.25 * 2 = 42.5 \pi \) units 3`
• For the cylinder:
  `\( V = \pi * 9 * 2.5 = 22.5 \pi \) units 3`
• For the hemisphere:
  `\( V = (2)/(3) \pi * 64 = 42.66 \pi \) units 3`
• For the sphere:
  `\( V = (4)/(3) \pi * 15.625 = 20.833 \pi \) units 3`

When comparing volumes, the one with the numerical smallest value will have the smallest volume, so the shape with the smallest computed volume is the sphere, option (D).