Question

A hemisphere with a slice removed is shown

the diagram below. If the radius of the sphe
4 meters, and the value of x is 60°, what is
volume of the figure to the nearest meter?
←x°-

Answer 1

112 m³

Explanation:

The given figure is a hemisphere with a slice removed, where the radius of the circular base of the hemisphere is 4 m, and the measure of the arc of the base of the removed slice is 60°.

Given that in a circle the measure of an arc is equal to the measure of its corresponding central angle, we can conclude that the central angle of the base of the removed slice is 60°.

Angles around a point sum to 360°. By dividing 60° by 360°, we find that the removed slice is equal to 1/6 of the entire hemisphere. Therefore, to find the volume of the figure, we need to find 5/6 of the volume of a hemisphere with radius 4 m.

`\boxed{\begin{array}{l}\underline{\textsf{Volume of a Hemisphere}}\\\\V=(2)/(3)\pi r^3\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\end{array}}`

Multiply the hemisphere formula by 5/6 and substitute r = 4:

`\begin{aligned}\textsf{Volume of the figure}&=(5)/(6) \cdot (2)/(3)\pi (4)^3\\\\&=(10)/(18)\pi \cdot 64\\\\&=(640)/(18)\pi\\\\&=111.7010721276...\\\\&=112\; \sf m^3\; (nearest\;meter)\end{aligned}\end{aligned}`

Therefore, the volume of the figure to the nearest meter is 112 m³.