Question

the large sphere has a diameter of 20 feet. which expression represents the volume, in cubic units, of the shaded part of the sphere? 4/3 π(10³) + 4/3 π(4³) 4/3 π(10³) − 4/3 π(4³) 4/3 π(20³) + 4/3 π(4³) 4/3 π(20³) − 4/3 π(4³)

Explanation:

Step1: Find the radius of the large sphere

The diameter of the large sphere is \(d = 20\) feet. The radius \(r=\frac{d}{2}\), so \(r = 10\) feet.

Step2: Recall the volume formula for a sphere

The volume formula for a sphere is \(V=\frac{4}{3}\pi r^{3}\).

Step3: Analyze the volume of the shaded part

The volume of the shaded part is the volume of the large sphere minus the volume of the small sphere.
The volume of the large sphere with radius \(r_1 = 10\) is \(V_1=\frac{4}{3}\pi(10)^{3}\).
The volume of the small sphere with radius \(r_2 = 4\) is \(V_2=\frac{4}{3}\pi(4)^{3}\).
So the volume of the shaded part \(V = V_1-V_2=\frac{4}{3}\pi(10)^{3}-\frac{4}{3}\pi(4)^{3}\)

Answer:

\(\frac{4}{3}\pi(10^{3})-\frac{4}{3}\pi(4^{3})\) (the second option)