Question

what is the volume of the solid generated by rotating a semicircle with a diameter of 6 mm about the axis? give your answer in terms of \\(\pi\\). enter your answer in the box. \\(\square\\) mm³

Explanation:

Step1: Identify the solid formed

When a semicircle is rotated about its diameter (the axis here), it forms a sphere. The diameter of the semicircle is the diameter of the sphere, so the diameter \( d = 6 \) mm, and the radius \( r=\frac{d}{2}=\frac{6}{2} = 3 \) mm.

Step2: Use the volume formula for a sphere

The volume \( V \) of a sphere is given by the formula \( V=\frac{4}{3}\pi r^{3} \). Substitute \( r = 3 \) mm into the formula:

$$ V=\frac{4}{3}\pi(3)^{3}=\frac{4}{3}\pi\times27 = 36\pi $$

Answer:

\( 36\pi \)