To solve for the volume of the sphere given the volume of the cone, we start by using the formula for the volume of a cone and the volume of a sphere to figure out the relationships between their dimensions.
-
Volume of the cone: The formula for the volume \( V \) of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \] where \( h \) is the height and \( r \) is the radius. In this case, we know that the volume of the cone is \( 253 \pi \) cmยณ.
Therefore: \[ \frac{1}{3} \pi r^2 h = 253 \pi \] Dividing both sides by \( \pi \): \[ \frac{1}{3} r^2 h = 253 \]
-
Height of the cone and diameter of the sphere: The problem states that the height of the cone \( h \) is equal to the diameter of the sphere. The diameter \( d \) of the sphere is related to the radius \( r \) of the sphere as follows: \[ d = 2r \] Thus, we have: \[ h = 2r \]
-
Substituting \( h \) into the cone volume equation: Now substitute \( h \) into the volume equation: \[ \frac{1}{3} r^2 (2r) = 253 \] Simplifying this gives: \[ \frac{2}{3} r^3 = 253 \] Multiplying both sides by 3: \[ 2 r^3 = 759 \] Dividing by 2: \[ r^3 = 379.5 \]
-
Calculating the volume of the sphere: The formula for the volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \]
Now substituting \( r^3 = 379.5 \) into the volume formula for the sphere: \[ V = \frac{4}{3} \pi (379.5) \]
-
Finding the volume: To find the approximate volume, calculate: \[ V \approx \frac{4 \times 379.5}{3} \pi = \frac{1518}{3} \pi = 506 \pi , \text{cm}^3 \]
Given that we rounded and approximated in the calculation, among the provided choices, the closest to our calculation is:
- 503 ฯ cmยณ
Thus, the volume of the sphere is approximately 503 ฯ cmยณ.