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A sphere of radius 2 inches is cut by three planes passing through its center. This partitions the solid into 8 equal parts, one of which is shown. The volume of each part is
t pi cubic inches. What is the value of
t ?

A sphere of radius $2$ inches is cut by three planes passing through its center. This partitions the solid into $8$ equal parts, one of which is shown. The volume of each part is $t \pi$ cubic inches. What is the value of $t$ ?

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Scale drawings: word problems

Full solution

Q. A sphere of radius

2

inches is cut by three planes passing through its center. This partitions the solid into

8

equal parts, one of which is shown. The volume of each part is

t \pi

cubic inches. What is the value of

t

?

Identify Formula for Volume: Identify the formula for the volume of a sphere. $\newline$ The volume $V$ of a sphere with radius $r$ is given by the formula $V = \frac{4}{3}\pi r^3$ .
Calculate Sphere Volume: Calculate the volume of the entire sphere using the given radius. $\newline$ The radius of the sphere is $2$ inches, so $V = \left(\frac{4}{3}\right)\pi(2 \text{ inches})^3$ . $\newline$ $V = \left(\frac{4}{3}\right)\pi(2^3)$ $\newline$ $V = \left(\frac{4}{3}\right)\pi(8)$ $\newline$ $V = \left(\frac{32}{3}\right)\pi$ cubic inches.
Determine Volume of One Part: Determine the volume of one of the eight equal parts. $\newline$ Since the sphere is cut into $8$ equal parts, the volume of each part is $\frac{1}{8}$ of the total volume. $\newline$ Volume of one part = $\left(\frac{1}{8}\right) * \left(\frac{32}{3}\right)\pi$ $\newline$ Volume of one part = $\left(\frac{32}{3}\right)\pi / 8$ $\newline$ Volume of one part = $\left(\frac{32}{3}\right)\pi * \left(\frac{1}{8}\right)$ $\newline$ Volume of one part = $\left(\frac{32}{24}\right)\pi$ $\newline$ Volume of one part = $\left(\frac{4}{3}\right)\pi$ cubic inches.
Compare Volume with Expression: Compare the volume of one part with the given volume expression. $\newline$ The volume of each part is given as $t\pi$ cubic inches. $\newline$ We found that the volume of one part is $(4/3)\pi$ cubic inches. $\newline$ Therefore, $t = 4/3$ .

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