Question

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

Answer 1

The volume of a cylinder with a circular cross-section area that is π/4 the area of the square cross-section of the square prism it fits inside is π/4 of the volume of the prism or πr²h. So the correct option is D.

Step-by-step explanation:

The student's question seems to be about finding the volume of a cylinder about the volume of a square prism in which it fits. Given that the area of the circle in any cross-section is π/4 the area of the square, the correct relationship for the volume of the cylinder compared to that of the square prism involves using the areas to determine the volumes.

The formula for the volume of a cylinder is V = πr²h, where 'r' is the radius and 'h' is the height. The volume of the square prism is the area of the square base ('s' being the side of the square) times the height, so V = (s²)h. Since the square is large enough to enclose the circular base of the cylinder, and each side 's' is equal to '2r', then the volume of the square prism is 4r²h. Hence, if the area of the circle is π/4 of the square area, the volume of the cylinder is also π/4 of the volume of the prism, which gives us the option (D) π/4 the volume of the prism or π/4(4r²)(h) or πr²h.

Answer 2

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