To solve this problem, we first need to find the radius of the beach ball and then calculate the volume of the cone that fits inside it.
Step 1: Find the radius of the beach ball
The diameter of the beach ball is given as 8 inches. The radius (r) can be found using the formula: \[ r = \frac{\text{diameter}}{2} = \frac{8 \text{ inches}}{2} = 4 \text{ inches} \]
Step 2: Determine the dimensions of the cone
The cone has the same radius as the beach ball, which is 4 inches, and a height of 8 inches.
Step 3: Calculate the volume of the cone
The formula for the volume (V) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] Where:
- \( r \) is the radius of the base of the cone
- \( h \) is the height of the cone
Substituting the values:
- \( r = 4 \) inches
- \( h = 8 \) inches
We get: \[ V = \frac{1}{3} \pi (4)^2 (8) \] \[ V = \frac{1}{3} \pi (16) (8) \] \[ V = \frac{1}{3} \pi (128) \] \[ V \approx \frac{128}{3} \pi \approx 134.04 \text{ in}^3 \quad (\text{using}\ \pi \approx 3.14) \]
Step 4: Calculate how many times the cone can fit into the beach ball
Next, we need to calculate the volume of the beach ball, which is a sphere. The formula for the volume (V) of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] Substituting the radius of the beach ball: \[ r = 4 \text{ inches} \] \[ V = \frac{4}{3} \pi (4)^3 \] \[ V = \frac{4}{3} \pi (64) \] \[ V = \frac{256}{3} \pi \approx 268.08 \text{ in}^3 \]
Finally, to find out how many times the cone fits into the beach ball: \[ \text{Number of cones} = \frac{\text{Volume of the beach ball}}{\text{Volume of the cone}} = \frac{\frac{256}{3} \pi}{\frac{128}{3} \pi} \] The \(\pi\) and the \(\frac{1}{3}\) cancel out: \[ \text{Number of cones} = \frac{256}{128} = 2 \]
Final Answers:
- The cone will fit into the beach ball 2 times.
- The volume of the cone is about 134.04 in³.
So, the final response is:
- The cone fits into the beach ball 2 times and the volume of the cone would be about 134.04 in³.