Question

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

Explanation:

Part (a)

Step1: Recall cone volume formula

The volume of a cone is given by \( V_{\text{cone}} = \frac{1}{3}Bh \), where \( B \) is the base area and \( h \) is the height.

Step2: Substitute values

Here, \( B = 11\ \text{ft}^2 \) and \( h = 3\ \text{ft} \). So, \( V_{\text{cone}} = \frac{1}{3} \times 11 \times 3 \).

Step3: Simplify

\( \frac{1}{3} \times 11 \times 3 = 11\ \text{ft}^3 \).

Part (b)

Step1: Recall cylinder volume formula

The volume of a cylinder is \( V_{\text{cylinder}} = Bh \), where \( B \) is the base area and \( h \) is the height.

Step2: Substitute values

With \( B = 11\ \text{ft}^2 \) and \( h = 3\ \text{ft} \), \( V_{\text{cylinder}} = 11 \times 3 \).

Step3: Calculate

\( 11 \times 3 = 33\ \text{ft}^3 \).

Part (c)

Step1: Find the ratio

From part (a) and (b), \( V_{\text{cone}} = 11 \) and \( V_{\text{cylinder}} = 33 \). So, \( \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{11}{33} = \frac{1}{3} \).

Step2: Analyze the statement

The formula \( V_{\text{cone}} = \frac{1}{3}Bh \) and \( V_{\text{cylinder}} = Bh \) hold when the cone and cylinder have the same base (congruent bases) and same height. So the equation \( V_{\text{cone}} = \frac{1}{3} \times V_{\text{cylinder}} \) is true for all cylinders and cones with congruent bases and equal heights.

Answer:

s:
(a) \( \boldsymbol{11} \)
(b) \( \boldsymbol{33} \)
(c) \( \boldsymbol{\frac{1}{3}} \); The correct option is: "This equation is true for all cylinders and cones with congruent bases and equal heights."