Match each description of a 3D solid to its corresponding numerical value for volume or to… — Armed Services Vocational Aptitude Battery (ASVAB)

Match each description of a 3D solid to its corresponding numerical value for volume or total surface area. (Note: All answers are expressed in terms of $\pi$ .)

Volume of a cylinder with a radius of $2$ and a height of $12$ $48\pi$
Volume of a sphere with a diameter of $6$ $36\pi$
Total surface area of a closed cylinder with a radius of $3$ and a height of $2$ $30\pi$
Surface area of a sphere with a radius of $4$ $64\pi$

Answer

The volume of the cylinder matches

48\pi

, the volume of the sphere matches

36\pi

, the total surface area of the cylinder matches

30\pi

, and the surface area of the sphere matches

64\pi

Each correct numerical value is found by identifying the specified 3D solid, recalling its corresponding volume or surface area formula, and accurately substituting the given dimensions into the formula.

Step-by-Step Solution

Calculate the volume of the cylinder.

V = \pi(2^2)(12) = 48\pi

Apply the cylinder volume formula

V = \pi r^2 h

Calculate the volume of the sphere.

Radius is

3

V = \frac{4}{3}\pi(3^3) = 36\pi

Convert diameter to radius, then apply the sphere volume formula

V = \frac{4}{3}\pi r^3

Calculate the total surface area of the closed cylinder.

SA = 2\pi(3^2) + 2\pi(3)(2) = 30\pi

Apply the cylinder total surface area formula

SA = 2\pi r^2 + 2\pi rh

Calculate the surface area of the sphere.

SA = 4\pi(4^2) = 64\pi

Apply the sphere surface area formula

SA = 4\pi r^2

Key Concept

Applying appropriate volume and surface area formulas for cylinders and spheres.

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