Cones, Cylinders, & Spheres Unit Test

1. Use the image to answer the question.

A cone is marked with a height of 15 centimeters and a diameter of 16 centimeters, both shown as perpendicular dotted lines. The slant height is 17 centimeters.

What is the height of the right circular cone?

(1 point)
Responses

17 cm
17 cm

16 cm
16 cm

8 cm
8 cm

15 cm

2. Use the image to answer the question.

An oblique cylinder is labeled with a height of 15 centimeters and radius 5 centimeters.

Find the surface area of the cylinder below. Use the approximation 3.14 for pi.

(1 point)
Responses

533.8 square centimeters
533.8 square centimeters

628 centimeters
628 centimeters

1,099 square centimeters
1,099 square centimeters

628 square centimeters
628 square centimeters

3. Adrian is painting the outside of a cylinder that he plans to use as a coffee table. The cylinder has a radius of 4 feet and a height of 3 feet. Adrian wants to paint all around the outside of the cylinder, including the top and bottom faces. In order to understand how much paint is needed, he wants to know the surface are of the outside of the cylinder. What is the surface area of the cylinder, measured in square feet? Use 3.14 for pi and round your answer to the nearest tenth.(1 point)
ft2

4. Eli is making a model castle out of clay. One of the roof peaks is in the shape of a cone with a diameter of 14 inches and a slant height of 20 inches. What is the surface area of the cone peak? Round your answer to the nearest hundredth. Use 3.14 for pi.(1 point)

5. Use the image to answer the question.

A 3 D cylinder shows a base radius of 8 millimeters and perpendicular height of 13 millimeters. A right angle is formed at the center of the base.

Find the volume of the cylinder, in cubic millimeters. Round your answer to the nearest hundredth.

(1 point)

6. A water bottle has a height of 16 inches and a radius of 4 inches. What is the volume, in cubic inches, of the water bottle? Use 3.14 for pi. (1 point)

7. Find the volume, in cubic inches, of a cone with a radius of 13 inches and a height of 27 inches. Round your answer to the nearest hundredth. Use 3.14 for pi.(1 point)

8. A cone-shaped container on a farm is used for storing animal feed. The container has a radius of 4 feet and a height of 12 feet. What volume of feed, in cubic feet, can the container store? Round your answer to the nearest hundredth. Use 3.14 for π
.(1 point)

9. The diameter of a sphere(1 point)
Responses

is the longest distance on a sphere.
is the longest distance on a sphere.

is the distance from the center of the sphere to any point on the surface of the sphere.
is the distance from the center of the sphere to any point on the surface of the sphere.

does not pass through the center of the sphere.
does not pass through the center of the sphere.

is half the length of the radius.

10. Find the surface area of a hemisphere with a radius 10m. Leave your answer in terms of pi.(1 point)
Responses

500π m2
500 pi m squared

300π m2
300 pi m squared

400π m2
400 pi m squared

200π m2

11. The surface area of a sphere is 7.84π ft.2 What is the radius of this sphere?(1 point)

12. How much ice cream is in a spherical scoop of ice cream with a radius of 3 cm? Use 3.14 as an approximation for pi. Round your answer to the nearest whole number.(1 point)

13. Suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point)
Responses

1 cylinder = 2 cones + 1 sphere
1 cylinder = 2 cones + 1 sphere

3 cones = 1 cylinder
3 cones = 1 cylinder

sphere = cylinder + cone
sphere = cylinder + cone

sphere - cone = cylinder

14. The volume of a sphere is 48 m3 . What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)

15. Use the image to answer the question.

A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.

Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.3 The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.

1 answer

Let's go through the questions one by one, providing the answers and calculations where necessary.

1. Height of the Right Circular Cone

  • Response: 15 cm
  • Explanation: The height of the cone is given directly as 15 cm.

2. Surface Area of an Oblique Cylinder

The formula for surface area \(S\) of a cylinder is given by: \[ S = 2 \pi r^2 + 2 \pi r h \] Where:

  • \( r = 5 \) cm (radius)
  • \( h = 15 \) cm (height)

Computing: \[ S = 2 \times 3.14 \times (5^2) + 2 \times 3.14 \times 5 \times 15 \] \[ S = 2 \times 3.14 \times 25 + 2 \times 3.14 \times 5 \times 15 \] \[ S = 157 + 471 \] \[ S = 628 \text{ square centimeters} \]

  • Response: 628 square centimeters

3. Surface Area of a Cylinder (Adrian's Coffee Table)

Using \( r = 4 \) feet and \( h = 3 \) feet: \[ S = 2 \pi r^2 + 2 \pi r h \] \[ S = 2 \times 3.14 \times (4^2) + 2 \times 3.14 \times 4 \times 3 \] \[ S = 2 \times 3.14 \times 16 + 2 \times 3.14 \times 12 \] \[ S = 100.48 + 75.36 \] \[ S = 175.84 \] Rounded to the nearest tenth: \[ S = 175.8 \text{ ft}^2 \]

4. Surface Area of a Cone (Castle Roof)

The formula for a cone's surface area is: \[ S = \pi r^2 + \pi r l \] Where:

  • Diameter = 14 inches → \( r = 7 \) inches
  • Slant height \( l = 20 \) inches

Computing: \[ S = \pi (7^2) + \pi (7)(20) \] \[ S = \pi (49) + \pi (140) \] \[ S = 189\pi \]

So surface area rounded to the nearest hundredth: \[ S \approx 593.86 \]

5. Volume of the Cylinder

Using \( r = 8 \) mm and \( h = 13 \) mm: \[ V = \pi r^2 h = \pi (8^2)(13) \] \[ V = \pi (64)(13) = 832\pi \] Approximation gives: \[ V \approx 2617.92 \text{ mm}^3 \]

6. Volume of the Water Bottle

Using \( r = 4 \) inches and \( h = 16 \) inches: \[ V = \pi r^2 h = \pi (4^2)(16) \] \[ V = \pi (16)(16) = 256\pi \] Approximately gives: \[ V \approx 804.25 \text{ in}^3 \]

7. Volume of Cone

Using \( r = 13 \) inches and \( h = 27 \) inches: \[ V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (13^2)(27) \] \[ V = \frac{1}{3}\pi (169)(27) = \frac{4563}{3}\pi = 1521\pi \] Approximated gives about \( 4777.89 \) cubic inches.

8. Volume of Cone-shaped Container

Using \( r = 4 \) feet, \( h = 12 \) feet:

  • Formula: \[ V = \frac{1}{3}\pi r^2 h \] Calculating: \[ V = \frac{1}{3} \pi (4^2)(12) = \frac{1}{3} \pi (16)(12) = \frac{1}{3} \pi (192) = 64\pi \] Approximation \( V \approx 201.06 \) cubic feet.

9. Diameter of a Sphere

  • Response: is the longest distance on a sphere.

10. Surface Area of a Hemisphere

Using \( r = 10 \) m: \[ S = 2\pi r^2 + \pi r^2 = 3\pi r^2 = 3\pi (10^2) = 300\pi \]

  • Response: 300π m²

11. Surface Area of a Sphere

Given \( 7.84\pi = 4\pi r^2 \): \[ r^2 = \frac{7.84}{4} = 1.96 \rightarrow r = \sqrt{1.96} \approx 1.4 \text{ ft} \]

12. Volume of Ice Cream Scoop

Using \( r = 3 \) cm: \[ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (3^3) = \frac{4}{3}\pi (27) = 36\pi \approx 113.04 \] Rounded to the nearest whole number: 113 cm³.

13. Relationship between Volumes

The correct relationship is:

  • Response: 1 cylinder = 2 cones + 1 sphere.

14. Volume of Cylinder

Assuming the sphere has volume \( 48 , \text{m}^3 \) and diameter matches the cylinder's height:

  • Use \( r \) from the sphere to find the cylinder \( V = \pi r^2 \cdot 2r = 2\pi r^3 \).

15. Total Volume in and on Top of Cone

If the ice cream cone's volume \( V = \frac{1}{3} \pi r^2 h \) and \( r = h \):

  • Calculate the total volume by using previously derived relationships (consider cone and hemisphere).

This provides a comprehensive review of these geometry problems. Let me know if you would like detailed calculations for any specific item or further clarification!

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