Task Sets: Composite 3D figures - ACT Math

A

Recall / Warm-Up

1.

What is the volume of a cylinder with radius 5 cm and height 8 cm? Leave your answer in terms of $\pi$ .

A.

$40\pi$ cm³

B.

$80\pi$ cm³

C.

$200\pi$ cm³

D.

$400\pi$ cm³

2.

What is the lateral surface area of a cylinder with radius 3 cm and height 10 cm?

A.

$30\pi$ cm²

B.

$60\pi$ cm²

C.

$78\pi$ cm²

D.

$90\pi$ cm²

3.

What is the volume of a hemisphere with radius 6 cm? Express your answer in terms of $\pi$ .

B

Fluency Practice

$A cone of height 4 m and radius 5 m sitting on top of a cylinder of height 12 m and radius 5 m, with labeled dimensions.$

1.

A solid is formed by placing a cone (radius 5 m, height 4 m) on top of a cylinder (radius 5 m, height 12 m). What is the total volume? Express your answer in terms of $\pi$ .

$A hemisphere of radius 4 cm resting on top of a cylinder of radius 4 cm and height 10 cm, with labeled dimensions.$

2.

A hemisphere of radius 4 cm is placed on top of a cylinder of radius 4 cm and height 10 cm. What is the total volume?

A.

$\frac{608\pi}{3}$ cm³

B.

$\frac{416\pi}{3}$ cm³

C.

$\frac{736\pi}{3}$ cm³

D.

$160\pi$ cm³

$A rectangular prism 12 cm by 8 cm by 8 cm with a cylindrical hole of radius 2 cm drilled through its full length.$

3.

A rectangular prism is 12 cm long, 8 cm wide, and 8 cm tall. A cylindrical hole with radius 2 cm is drilled through the entire 12 cm length. What is the remaining volume in cubic centimeters? Round to the nearest whole number.

$A hemisphere of radius 4 cm on top of a cylinder of radius 4 cm and height 10 cm, with exterior surfaces highlighted and the hidden contact areas at the joint shown in gray.$

4.

A hemisphere of radius 4 cm sits on top of a cylinder of radius 4 cm and height 10 cm. What is the total exterior surface area?

A.

$128\pi$ cm²

B.

$160\pi$ cm²

C.

$144\pi$ cm²

D.

$112\pi$ cm²

$A cone of radius 3 cm and slant height 5 cm on top of a cylinder of radius 3 cm and height 7 cm, with dimensions labeled.$

5.

A cone with radius 3 cm and slant height 5 cm sits on top of a cylinder with radius 3 cm and height 7 cm. What is the total exterior surface area? Express your answer in terms of $\pi$ .

C

Varied Practice

D

Word Problems / Application

$A grain silo consisting of a cylinder of radius 4 m and height 10 m with a conical roof of height 3 m on top.$

1.

A grain silo has a cylindrical body with radius 4 m and height 10 m, topped with a conical roof with the same radius and a height of 3 m.

What is the total volume of the silo? Express your answer in terms of $\pi$ .

$A horizontal storage tank: cylinder of radius 3 m and length 8 m with a hemispherical cap of radius 3 m on each end.$

2.

A storage tank consists of a cylinder with radius 3 m and height 8 m, with a hemisphere attached to each end.

1.

What is the total volume of the tank? Express your answer in terms of $\pi$ .

2.

What is the total exterior surface area of the tank? Express your answer in terms of $\pi$ .

E

Error Analysis

1.

Priya solved this problem:

"A hemisphere of radius 5 cm sits on top of a cylinder of radius 5 cm and height 8 cm. Find the total exterior surface area."

Priya's work:

Cylinder total SA: $2\pi(5)(8) + 2\pi(5)^2 = 80\pi + 50\pi = 130\pi$
Hemisphere total SA: $2\pi(5)^2 + \pi(5)^2 = 50\pi + 25\pi = 75\pi$
Total: $130\pi + 75\pi = 205\pi$ cm²

What error did Priya make, and what is the correct surface area?

A.

Priya included the hidden contact circles in both shapes. The correct answer is $155\pi$ cm².

B.

Priya subtracted the contact area only once. The correct answer is $180\pi$ cm².

C.

Priya used the wrong hemisphere curved SA formula. The correct answer is $130\pi$ cm².

D.

Priya's answer of $205\pi$ cm² is correct.

2.

Marcus solved this problem:

"A hemisphere of radius 6 cm sits on top of a cylinder of radius 6 cm and height 5 cm. Find the total volume."

Marcus's work:

Cylinder: $V = \pi(6)^2(5) = 180\pi$
Hemisphere: $V = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi$
Total: $180\pi + 288\pi = 468\pi$ cm³

What error did Marcus make, and what is the correct volume?

A.

Marcus used the full sphere formula instead of the hemisphere formula. The correct volume is $324\pi$ cm³.

B.

Marcus should have subtracted the hemisphere volume. The correct volume is $36\pi$ cm³.

C.

Marcus forgot to subtract the contact area. The correct volume is $432\pi$ cm³.

D.

Marcus's answer of $468\pi$ cm³ is correct.

F

Challenge / Extension

$A trophy made of three stacked parts: a cylinder (r = 4 cm, h = 6 cm) at the bottom, a cone (h = 9 cm) in the middle, and a hemisphere on top, all with the same radius.$

1.

A trophy is made of three parts stacked vertically: a cylinder (radius 4 cm, height 6 cm) on the bottom, a cone (radius 4 cm, height 9 cm) in the middle, and a hemisphere (radius 4 cm) on top. What is the total volume? Express your answer in terms of $\pi$ .

2.