Question

If the diameter of a small red beach ball is 8 inches, then the cone with the same radius and a height of 8 inches would fit into the beach ball (Response area) times and the volume of the cone would be about . (Response area) in^3.

Answer 1

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**.