Help with calculus please

Question

The base of a solid in the xy-plane is the circle x² + y² = 16. Cross sections of the solid perpendicular to the y-axis are semicircles. What is the volume, in cubic units, of the solid?

Options:

1)
2)
3)
4)

Answer 1

Correct answer is 1)

If the base is a circle and perpendicular cross-sections are semi-circles, the solid could only be a hemisphere. The radius of the hemisphere is 4, according to the equation of the circle at its base. The volume of a hemisphere is given by:

V = 2/3*pi*R³, where R is the radius of the hemisphere. Plugging in the R = 4, we obtain:

V = 2/3*pi*4³ = 128pi/3