Question

card a
determine the volume of the given solid.

card b
determine the volume of the given solid.

card c
determine the volume of the given solid.

card d
determine the volume of the given solid.

card e
determine the volume of the given solid.

card f
determine the volume of the given solid.

card g
determine the volume of the given solid.

card h
determine the volume of the given solid.

card i
determine the volume of the cone that has a base and height that is congruent to this cylinders base and height.

card j
determine the volume of the cone that has a base and height that is congruent to this cylinders base and height.

card k
determine the volume of the cylinder that has a base and height that is congruent to this cones base and height.

a-maze-ing volume questions

Explanation:

Volume of Cylinders (Cards A, B, C, D)

Using the Volume of Cylinders knowledge point

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Volume of Cones (Cards E, F, G, H)

The volume of a cone is given by the formula:

$$V = \frac{1}{3}\pi r^2 h$$

Let's calculate the volume for each cone:

• Card E: The radius \(r = 2\text{ m}\) and the height \(h = 5\text{ m}\).

$$V = \frac{1}{3}\pi (2)^2 (5) = \frac{20}{3}\pi \approx 20.94\text{ m}^3$$

• Card F: The radius \(r = 1\text{ ft}\) and the height \(h = 5\text{ ft}\).

$$V = \frac{1}{3}\pi (1)^2 (5) = \frac{5}{3}\pi \approx 5.24\text{ ft}^3$$

• Card G: The radius \(r = 4\text{ ft}\) and the height \(h = 6\text{ ft}\).

$$V = \frac{1}{3}\pi (4)^2 (6) = 32\pi \approx 100.53\text{ ft}^3$$

• Card H: The radius \(r = 2\text{ mi}\) and the height \(h = 2\text{ mi}\).

$$V = \frac{1}{3}\pi (2)^2 (2) = \frac{8}{3}\pi \approx 8.38\text{ mi}^3$$

Cone and Cylinder Relationships (Cards I, J, K)

We apply the relationship between the volume of a cylinder (\(V_{\text{cylinder}} = \pi r^2 h\)) and a cone (\(V_{\text{cone}} = \frac{1}{3}\pi r^2 h\)) with congruent bases and heights:

• Card I: We need the volume of a cone with a base and height congruent to the given cylinder (\(r = 5\text{ mi}\), \(h = 16\text{ mi}\)).

$$V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3}\pi (5)^2 (16) = \frac{400}{3}\pi \approx 418.88\text{ mi}^3$$

• Card J: We need the volume of a cone with a base and height congruent to the given cylinder (\(r = 5\text{ ft}\), \(h = 8\text{ ft}\)).

$$V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3}\pi (5)^2 (8) = \frac{200}{3}\pi \approx 209.44\text{ ft}^3$$

• Card K: We need the volume of a cylinder with a base and height congruent to the given cone (\(r = 2\text{ km}\), \(h = 4\text{ km}\)).

$$V_{\text{cylinder}} = 3 V_{\text{cone}} = \pi r^2 h = \pi (2)^2 (4) = 16\pi \approx 50.27\text{ km}^3$$

Answer:

Card	Solid Type	Formula Applied	Exact Volume (in terms of \(\pi\))	Approximate Volume (to 2 decimal places)
Card B	Cylinder	\(V = \pi r^2 h\)	\(216\pi\text{ cm}^3\)	\(678.58\text{ cm}^3\)
Card C	Cylinder	\(V = \pi r^2 h\)	\(45\pi\text{ cm}^3\)	\(141.37\text{ cm}^3\)
Card D	Cylinder	\(V = \pi r^2 h\)	\(180\pi\text{ m}^3\)	\(565.49\text{ m}^3\)
Card E	Cone	\(V = \frac{1}{3}\pi r^2 h\)	\(\frac{20}{3}\pi\text{ m}^3\)	\(20.94\text{ m}^3\)
Card F	Cone	\(V = \frac{1}{3}\pi r^2 h\)	\(\frac{5}{3}\pi\text{ ft}^3\)	\(5.24\text{ ft}^3\)
Card G	Cone	\(V = \frac{1}{3}\pi r^2 h\)	\(32\pi\text{ ft}^3\)	\(100.53\text{ ft}^3\)
Card H	Cone	\(V = \frac{1}{3}\pi r^2 h\)	\(\frac{8}{3}\pi\text{ mi}^3\)	\(8.38\text{ mi}^3\)
Card I	Cone	\(V = \frac{1}{3}\pi r^2 h\)	\(\frac{400}{3}\pi\text{ mi}^3\)	\(418.88\text{ mi}^3\)
Card J	Cone	\(V = \frac{1}{3}\pi r^2 h\)	\(\frac{200}{3}\pi\text{ ft}^3\)	\(209.44\text{ ft}^3\)
Card K	Cylinder	\(V = \pi r^2 h\)	\(16\pi\text{ km}^3\)	\(50.27\text{ km}^3\)