Question
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 256 π cm3 Start Fraction 25 over 6 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 252 π cm3
Answers
Answered by
GPT-4o mini
To find the volume of the sphere based on the given information about the cone, we will start with the formula for the volume of a cone and the relationship between the cone and the sphere.
1. **Volume of the cone**: The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius of the cone,
- \( h \) is the height of the cone.
We know that the volume of the cone is \( 253 \pi \, \text{cm}^3 \). Therefore:
\[
\frac{1}{3} \pi r^2 h = 253 \pi
\]
Dividing both sides by \( \pi \):
\[
\frac{1}{3} r^2 h = 253
\]
Multiplying by 3:
\[
r^2 h = 759
\]
2. **Relationship with the sphere**: The height of the cone \( h \) is equal to the diameter of the sphere. Since the diameter of the sphere is \( 2r \), we have:
\[
h = 2r
\]
Substituting \( h = 2r \) into the equation \( r^2 h = 759 \):
\[
r^2 (2r) = 759
\]
This simplifies to:
\[
2r^3 = 759 \implies r^3 = \frac{759}{2} = 379.5
\]
3. **Volume of the sphere**: The volume \( V_s \) of a sphere is given by:
\[
V_s = \frac{4}{3} \pi r^3
\]
Now, substituting \( r^3 = 379.5 \):
\[
V_s = \frac{4}{3} \pi (379.5)
\]
Calculating \( \frac{4}{3} \times 379.5 \):
\[
\frac{4 \times 379.5}{3} = \frac{1518}{3} = 506
\]
Therefore:
\[
V_s = 506 \pi
\]
So, the volume of the sphere is \( 506 \pi \, \text{cm}^3 \).
**Final Answer**: \( 503 \pi \, \text{cm}^3 \) (this answer seems closest given the options may have a typing error in the original question).
1. **Volume of the cone**: The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius of the cone,
- \( h \) is the height of the cone.
We know that the volume of the cone is \( 253 \pi \, \text{cm}^3 \). Therefore:
\[
\frac{1}{3} \pi r^2 h = 253 \pi
\]
Dividing both sides by \( \pi \):
\[
\frac{1}{3} r^2 h = 253
\]
Multiplying by 3:
\[
r^2 h = 759
\]
2. **Relationship with the sphere**: The height of the cone \( h \) is equal to the diameter of the sphere. Since the diameter of the sphere is \( 2r \), we have:
\[
h = 2r
\]
Substituting \( h = 2r \) into the equation \( r^2 h = 759 \):
\[
r^2 (2r) = 759
\]
This simplifies to:
\[
2r^3 = 759 \implies r^3 = \frac{759}{2} = 379.5
\]
3. **Volume of the sphere**: The volume \( V_s \) of a sphere is given by:
\[
V_s = \frac{4}{3} \pi r^3
\]
Now, substituting \( r^3 = 379.5 \):
\[
V_s = \frac{4}{3} \pi (379.5)
\]
Calculating \( \frac{4}{3} \times 379.5 \):
\[
\frac{4 \times 379.5}{3} = \frac{1518}{3} = 506
\]
Therefore:
\[
V_s = 506 \pi
\]
So, the volume of the sphere is \( 506 \pi \, \text{cm}^3 \).
**Final Answer**: \( 503 \pi \, \text{cm}^3 \) (this answer seems closest given the options may have a typing error in the original question).