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