Question

A sphere fits snugly inside a 6-in. cube as shown. what is the volume of the region inside the cube but outside the sphere

Answer 1

The volume of the region inside the cube but outside the sphere is approximately 187.73 in^3.

Step-by-step explanation:

The volume of the region inside the cube but outside the sphere can be calculated by subtracting the volume of the sphere from the volume of the cube.

Volume of the cube = side^3 = 6^3 = 216 in^3

Volume of the sphere = (4/3)πr^3

Since the sphere fits snugly inside the cube, the radius of the sphere is half the length of the side of the cube. So, r = 3/2 in.

Therefore, volume of the region = volume of the cube - volume of the sphere = 216 in^3 - (4/3)π(3/2)^3 in^3 = 216 in^3 - 9π in^3 = 216 in^3 - 28.27 in^3 ≈ 187.73 in^3.

Answer 2

so hmm as shown below, check the picture.

so, simply, get the volume of the cube, length * height * width, is a 6x6x6. and then get the volume of the sphere, and the volume of the cube "includes" the volume of the sphere, however, if you subtract the volume of the sphere, what's leftover, is the part that's outside the sphere, and inside the cube.


`\bf \textit{volume of a cube}\\\\ V=lwh\quad \begin{cases} l=length\\ w=width\\ h=height\\ ------\\ l=w=h=6 \end{cases}\implies V=6^3 \\\\\\ \textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}\quad \begin{cases} r=radius\\ -----\\ r=3 \end{cases}\implies V=\cfrac{4\pi \cdot 3^3}{3}`