To solve the problem, we first need to find the radius of the cone since its volume is given to us.
The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height of the cone.
We know from the problem that: \[ V = 253 \pi \] So we can express it as: \[ \frac{1}{3} \pi r^2 h = 253 \pi \] Dividing both sides by \( \pi \): \[ \frac{1}{3} r^2 h = 253 \] Multiplying both sides by 3: \[ r^2 h = 759 \]
Next, we need to relate the height of the cone to the diameter of the sphere. According to the problem, the height of the cone equals the diameter of the sphere. The diameter \( d \) of the sphere is twice the radius: \[ d = 2r \] Thus, we can say: \[ h = 2r \] Substituting \( h \) in our earlier equation: \[ r^2 (2r) = 759 \] This simplifies to: \[ 2r^3 = 759 \] Dividing both sides by 2: \[ r^3 = \frac{759}{2} = 379.5 \]
Now we can find the radius: \[ r = \sqrt[3]{379.5} \]
Once we have \( r \), we need to find the volume \( V \) of the sphere. The volume \( V_s \) of a sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 = 379.5 \): \[ V_s = \frac{4}{3} \pi (379.5) \] Calculating this gives: \[ V_s = \frac{4 \times 379.5}{3} \pi = \frac{1518}{3} \pi \] Thus: \[ V_s = 506 \pi \]
Looking at the responses provided:
- \( 503 \pi , \text{cm}^3 \)
- \( \frac{50}{3} \pi , \text{cm}^3 \) (approximately \( 16.67\pi , \text{cm}^3 \))
- other values do not correspond.
It seems we calculated with a slight error, as \( r^3 \) should match provided multiple parameters; however, none of the options match the calculated \( 506 \pi , \text{cm}^3 \) as closely fitting; thus we should round or reconsider based on the error tolerance.
The answer that comes closest based on typical expectations on bounds could be the \( 503 \pi \). Thus, based interpreting standard limits:
Response: \( 503 \pi , \text{cm}^3 \).
1 answer