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 \\(\frac{\pi r^2}{4r^2}\\) or \\(\frac{\pi}{4}\\). cross section since the area of the circle is \\(\frac{\pi}{4}\\) the area of the square, the volume of the cylinder equals \\(\bigcirc\\) \\(\frac{\pi}{2}\\) the volume of the prism or \\(\frac{\pi}{2}(2r)(h)\\) or \\(\pi rh\\). \\(\bigcirc\\) \\(\frac{\pi}{2}\\) the volume of the prism or \\(\frac{\pi}{2}(4r^2)(h)\\) or \\(2\pi rh\\). \\(\bigcirc\\) \\(\frac{\pi}{4}\\) the volume of the prism or \\(\frac{\pi}{4}(2r)(h)\\) or \\(\frac{\pi}{4}r^2h\\). \\(\bigcirc\\) \\(\frac{\pi}{4}\\) the volume of the prism or \\(\frac{\pi}{4}(4r^2)(h)\\) or \\(\pi r^2h\\).

Explanation:

Step1: Recall Volume Formulas

The volume of a square prism (a rectangular prism with square base) is \( V_{\text{prism}}=\text{base area}\times\text{height} \). The base of the square prism is a square with side length \( 2r \), so base area \( A_{\text{square}}=(2r)^2 = 4r^2 \), and height is \( h \), so \( V_{\text{prism}}=4r^2h \).

The volume of a cylinder is \( V_{\text{cylinder}}=\pi r^2h \), where \( r \) is the radius of the base of the cylinder and \( h \) is the height (same as the prism's height since the cylinder fits inside).

Step2: Relate Cylinder and Prism Volumes

We know the ratio of the cross - sectional area (circle to square) is \( \frac{\pi}{4} \). Since volume of a prism (or cylinder) is base area times height, and the height \( h \) is the same for both the cylinder and the prism, the ratio of their volumes will be the same as the ratio of their base areas.

So \( V_{\text{cylinder}}=\frac{\pi}{4}V_{\text{prism}} \). Substituting \( V_{\text{prism}} = 4r^2h \) into this formula, we get \( V_{\text{cylinder}}=\frac{\pi}{4}(4r^2h)=\pi r^2h \), which is the formula for the volume of a cylinder.

Answer:

\(\frac{\pi}{4}\) the volume of the prism or \(\frac{\pi}{4}(4r^{2})(h)\) or \(\pi r^{2}h\) (the last option among the given choices)