Question

the cylinder and cone shown below have congruent bases and equal heights.
complete the following.
(a) volume of the cylinder: \\(\square\\,\text{m}^3\\)
(b) volume of the cone: \\(\square\\,\text{m}^3\\)
(c) volume of the cylinder = \\(\square \times \\) volume of the cone
\\(\circ\\) this equation is true for all cylinders and cones.
\\(\circ\\) this equation is true for all cylinders and cones with congruent bases and equal heights.
\\(\circ\\) this equation is true only for the cylinder and cone shown above.

Explanation:

Part (a)

Step1: Recall cylinder volume formula

The volume of a cylinder is given by \( V = \text{Base Area} \times \text{Height} \).

Step2: Identify base area and height

The base area (\( B \)) is \( 15 \, \text{m}^2 \) and the height (\( h \)) is \( 5 \, \text{m} \).

Step3: Calculate volume

\( V_{\text{cylinder}} = 15 \times 5 = 75 \, \text{m}^3 \).

Step1: Recall cone volume formula

The volume of a cone is given by \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

Step2: Identify base area and height

The base area (\( B \)) is \( 15 \, \text{m}^2 \) and the height (\( h \)) is \( 5 \, \text{m} \).

Step3: Calculate volume

\( V_{\text{cone}} = \frac{1}{3} \times 15 \times 5 = 25 \, \text{m}^3 \).

Step1: Find the ratio of cylinder to cone volume

From part (a) and (b), \( \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{75}{25} = 3 \).

Step2: Analyze the statement

The formula for the volume of a cylinder is \( V_{\text{cylinder}} = B \times h \) and for a cone is \( V_{\text{cone}} = \frac{1}{3} B \times h \) (when they have the same base area \( B \) and height \( h \)). So, for any cylinder and cone with congruent bases (same base area) and equal heights, \( V_{\text{cylinder}} = 3 \times V_{\text{cone}} \).

Answer:

\( 75 \)

Part (b)