A cylinder fits inside a square prism as shown. For every cross section, the ratio of the area of the circle to the area of the square is or nr^2 or n/4. Since the area of the circle is n/4 the area of the square, the volume of the cylinder equals

A) pi/2 the volume of the prism or pi/2 (2r) (h) or pi*r*h

B) pi/2 the volume of the prism or pi/2 (4r^2) (h) or 2*pi*r*h.

C) pi/4 the volume of the prism or pi/4 (2r) (h) or pi/4*r^2*h.

D) pi/4 the volume of the prism or pi/4 (4r^2) (h) or pi*r^2*h

Area circle = π*r²

Area square = l²

The side of the square is equal to the diameter of the circle

Area square = D²

A diameter is always twice the radius

Area square = (2r) ² = 2²r² = 4r²

So this is the rate:

Area circle/Area square = (π*r²) / (4r²)

Area circle/Area square = π/4

Volume is always Area*h when cross sectional area is a constant

Volume Prism = Area Square*h

Volume Prism = 4r²*h

Volume Cylinder = Area Circle*h

Volume Cylinder = π*r²*h

So far this is option D)

Let’s calculate the rate:

Volume Cylinder/Volume Prism = π*r²*h/4r²*h

Volume Cylinder/Volume Prism = π/4

Volume Cylinder = π/4 * Volume Prism

This is also option D)

Now let’s calculate Volume Cylinder from that formula:

Volume Cylinder = π/4 * Volume Prism

Volume Cylinder = π/4 * (4r²*h)

This is also option D)

So option D) is correct