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 cone = \\(\square \times \\) volume of the cylinder
\\(\bigcirc\\) this equation is true for all cylinders and cones with congruent bases and equal heights.
\\(\bigcirc\\) this equation is true only for the cylinder and cone shown above.
\\(\bigcirc\\) this equation is true for all cylinders and cones.

Explanation:

Part (a) - Volume of the Cylinder

Step 1: Recall the formula for the volume of a cylinder

The volume \( V \) of a cylinder is given by \( V = \text{Base Area} \times \text{Height} \). The base is a circle, so the base area \( A = \pi r^2 \), but here we are given the base area as \( 18 \, \text{m}^2 \) (wait, actually, looking at the diagram, the base area is labeled as \( 18 \, \text{m}^2 \)? Wait, no, maybe that's a typo? Wait, no, the cylinder has height 7 m, and the base area (the area of the circular base) is... Wait, maybe the \( 18 \, \text{m}^2 \) is the base area? Wait, no, let's check again. Wait, the problem says "congruent bases", so the base area of the cylinder and cone is the same. Wait, the diagram shows for the cylinder, the base (the circular face) has an area of \( 18 \, \text{m}^2 \)? Wait, no, maybe that's the area? Wait, no, maybe it's the radius? Wait, no, the label is \( 18 \, \text{m}^2 \), so that's the area. Wait, no, that doesn't make sense. Wait, maybe it's a mistake, and actually, the base area is \( 18 \, \text{m}^2 \)? Wait, no, let's re-examine. Wait, the formula for the volume of a cylinder is \( V = \text{Base Area} \times \text{Height} \). The height of the cylinder is 7 m, and the base area (the area of the circular base) is given as \( 18 \, \text{m}^2 \)? Wait, no, maybe the \( 18 \, \text{m}^2 \) is the area. Wait, let's proceed. If the base area \( A = 18 \, \text{m}^2 \) and height \( h = 7 \, \text{m} \), then the volume of the cylinder is \( V_{\text{cylinder}} = A \times h \).

Step 2: Calculate the volume of the cylinder

So, \( V_{\text{cylinder}} = 18 \, \text{m}^2 \times 7 \, \text{m} = 126 \, \text{m}^3 \)? Wait, no, that can't be right. Wait, maybe the \( 18 \, \text{m}^2 \) is a typo, and it's the radius? Wait, no, the label is \( 18 \, \text{m}^2 \), so area. Wait, but let's check the cone. The cone has height 7 m, same as the cylinder, and base area 18 m². Then the volume of the cone is \( \frac{1}{3} \times \text{Base Area} \times \text{Height} \). So maybe the base area is 18 m². So for the cylinder, volume is base area times height: \( 18 \times 7 = 126 \)? Wait, but that seems too small. Wait, maybe the \( 18 \, \text{m}^2 \) is actually the area of the base, so let's go with that. Wait, but let's confirm. Wait, the problem says "congruent bases and equal heights". So base area of cylinder = base area of cone = 18 m², height of cylinder = height of cone = 7 m. Then:

Volume of cylinder: \( V_{\text{cylinder}} = \text{Base Area} \times \text{Height} = 18 \, \text{m}^2 \times 7 \, \text{m} = 126 \, \text{m}^3 \)? Wait, no, that seems low. Wait, maybe the \( 18 \, \text{m}^2 \) is a mistake, and it's the radius? Wait, no, the unit is m², so it's area. Wait, maybe the diagram has a typo, and the base area is actually \( 18 \pi \, \text{m}^2 \)? No, the problem shows \( 18 \, \text{m}^2 \). Let's proceed with the given numbers.

So, Volume of cylinder: \( 18 \times 7 = 126 \, \text{m}^3 \).

Part (b) - Volume of the Cone

Step 1: Recall the formula for the volume of a cone

The volume \( V \) of a cone is given by \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). The base area is the same as the cylinder's base area, which is \( 18 \, \text{m}^2 \), and the height is 7 m (same as the cylinder).

Step 2: Calculate the volume of the cone

So, \( V_{\text{cone}} = \frac{1}{3} \times 18 \, \text{m}^2 \times 7 \, \text{m} \). First, \( \frac{1}{3} \times 18 = 6 \), then \( 6 \times 7 = 42 \, \text{m}^3 \).

Part (c) - Relationship between the Volumes

Step 1: Find the ratio of the cone's volume to the cylinder's volume

We have \( V_{\text{cone}} = 42 \, \text{m}^3 \) and \( V_{\text{cylinder}} = 126 \, \text{m}^3 \). So the ratio is \( \frac{42}{126} = \frac{1}{3} \).

Step 2: Determine the validity of the equation

The formula for the volume of a cone is \( \frac{1}{3} \times \text{Base Area} \times \text{Height} \), and the volume of a cylinder is \( \text{Base Area} \times \text{Height} \). So if a cylinder and cone have congruent bases (same base area) and equal heights, then the volume of the cone is always \( \frac{1}{3} \) the volume of the cylinder. So the first option is correct: "This equation is true for all cylinders and cones with congruent bases and equal heights."

Now, let's summarize:

(a) Volume of the cylinder: \( 18 \times 7 = 126 \, \text{m}^3 \)

(b) Volume of the cone: \( \frac{1}{3} \times 18 \times 7 = 42 \, \text{m}^3 \)

(c) The ratio is \( \frac{1}{3} \), and the correct statement is the first one.

Answer:

s:

(a) \( \boxed{126} \)

(b) \( \boxed{42} \)

(c) The value is \( \boxed{\frac{1}{3}} \), and the correct option is "This equation is true for all cylinders and cones with congruent bases and equal heights."