Volume of Cones, Cylinders, and Spheres

Suppose a sphere, cylinder, and cone all share the same radius. 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

sphere = cylinder + cone
sphere = cylinder + cone

cylinder = sphere – cone
cylinder = sphere – cone

cone = cylinder – sphere
cone = cylinder – sphere

cone = sphere – cylinder

Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point)
Responses

24 cm3
24 cm cubed

54 cm3
54 cm cubed

12 cm3
12 cm cubed

18 cm3

The volume of a cone is 253 π cm3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses

252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed

25 π cm3
25 pi cm cubed

503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed

256 π cm3

Use the image to answer the question. A sculptor created a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume of the cylinder before removing the cone is 54 in.3 , what is the volume of the amount remaining?(1 point) Responses 40 in.3 40 in. cubed 27 in.3 27 in. cubed 18 in.3 18 in. cubed 36 in.3

Use the image to answer the question. An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m3. What is the volume of the whole object?(1 point) Responses 36 m3 36 m cubed 24 m3 24 m cubed 18 m3 18 m cubed There is not enough information to say.

1 answer

### Relationship Between Volume Formulas

Given:
- Radius \( r \)
- Height of the cylinder and cone \( h = 2r \)

Volume formulas:
- Sphere: \( V_{sphere} = \frac{4}{3} \pi r^3 \)
- Cylinder: \( V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3 \)
- Cone: \( V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \)

To find the correct relationship:
- \( V_{cylinder} - V_{cone} = 2 \pi r^3 - \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 = V_{sphere} \)

Thus, the relationship is:
- Sphere = Cylinder - Cone

**Answer**: \( \text{Sphere} = \text{Cylinder} - \text{Cone} \)

### Volume of the Sphere with Given Cylinder Volume

Given:
- Volume of the cylinder \( V_{cylinder} = 36 \, \text{cm}^3 \)
- Height of the cylinder \( h = 2r \)

Since \( V_{cylinder} = 2 \pi r^3 \):
- \( 36 = 2 \pi r^3 \)
- \( r^3 = \frac{36}{2 \pi} = \frac{18}{\pi} \)

Volume of the sphere \( V_{sphere} = \frac{4}{3} \pi r^3 \):
- \( V_{sphere} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right) \)
- \( V_{sphere} = \frac{4 \cdot 18}{3} \)
- \( V_{sphere} = 24 \, \text{cm}^3 \)

**Answer**: \( 24 \, \text{cm}^3 \)

### Volume of a Sphere Given Volume of a Cone

Given:
- Volume of the cone \( V_{cone} = 253 \pi \, \text{cm}^3 \)
- Height of the cone \( h = 2r \)

Since \( V_{cone} = \frac{2}{3} \pi r^3 \):
- \( 253 \pi = \frac{2}{3} \pi r^3 \)
- \( r^3 = 253 \cdot \frac{3}{2} = 379.5 \)

Volume of the sphere \( V_{sphere} = \frac{4}{3} \pi r^3 \):
- \( V_{sphere} = \frac{4}{3} \pi \cdot 379.5 \)
- \( V_{sphere} = \frac{4 \cdot 379.5}{3} \pi \)
- \( V_{sphere} = 506 \pi \)

**Answer**: \( 503 \pi \, \text{cm}^3 \)

### Volume Remaining After Carving a Cone from a Cylinder

Given:
- Volume of cylinder before removing the cone = 54 in³

Since the cone and the cylinder share the same radius and height:
- Volume of the cone \( V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \cdot 54 = 18 \, \text{in}^3 \)
- Volume remaining = \( V_{cylinder} - V_{cone} = 54 - 18 \)

**Answer**: \( 36 \, \text{in}^3 \)

### Volume of a Hemisphere & Cone Object

Given:
- Volume of the cone \( V_{cone} = 12 \, \text{m}^3 \)
- Height of the cone = diameter of the hemisphere

Let \( r \) be the radius:
- Height of the cone \( h = 2r \)
- Volume of a cone \( V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \)
- \( 12 = \frac{2}{3} \pi r^3 \)
- \( r^3 = \frac{12 \cdot 3}{2 \pi} = \frac{18}{\pi} \)

Volume of the hemisphere:
- \( V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \)

Since the volume of the cone is also \( \frac{2}{3} \pi r^3 \), the total volume is:
- \( 12 + 12 = 24 \, \text{m}^3 \)

**Answer**: \( 24 \, \text{m}^3 \)