Question
Volume of a container measures 3200 in and contains a dozen tennis balls each ball has a radius if 2.5 in how much space in the container is empty use 3.14 for pi and round the nearest answer whole number
Answers
GPT-4o mini
To find the empty space in the container after accounting for the volume of the tennis balls, we first need to calculate the volume of one tennis ball and then the total volume of all twelve tennis balls.
The formula for the volume \( V \) of a sphere (which is the shape of a tennis ball) is given by:
\[
V = \frac{4}{3} \pi r^3
\]
Where \( r \) is the radius of the sphere.
1. **Calculate the volume of one tennis ball:**
- Given the radius \( r = 2.5 \) inches, we can substitute this value into the formula. Using \( \pi \approx 3.14 \):
\[
V = \frac{4}{3} \cdot 3.14 \cdot (2.5)^3
\]
First, calculate \( (2.5)^3 \):
\[
(2.5)^3 = 15.625
\]
Now plug this value back into the volume formula:
\[
V = \frac{4}{3} \cdot 3.14 \cdot 15.625
= \frac{4 \cdot 3.14 \cdot 15.625}{3}
\]
Calculate \( 4 \cdot 3.14 \):
\[
4 \cdot 3.14 = 12.56
\]
Now multiply by 15.625:
\[
12.56 \cdot 15.625 \approx 196.09375
\]
Finally, divide by 3:
\[
V \approx \frac{196.09375}{3} \approx 65.36458333
\]
So the volume of one tennis ball is approximately \( 65.36 \) cubic inches.
2. **Calculate the total volume for a dozen (12) tennis balls:**
\[
V_{\text{total}} = 12 \cdot V \approx 12 \cdot 65.36 \approx 784.32 \text{ cubic inches}
\]
3. **Calculate the empty space in the container:**
The volume of the container is given as \( 3200 \) cubic inches. To find the empty space, subtract the total volume of the tennis balls from the volume of the container:
\[
\text{Empty space} = \text{Volume of container} - \text{Volume of tennis balls}
\]
\[
\text{Empty space} = 3200 - 784.32 \approx 2415.68
\]
4. **Round the empty space to the nearest whole number:**
\[
\text{Empty space} \approx 2416 \text{ cubic inches}
\]
Thus, the amount of empty space in the container is approximately **2416 cubic inches**.
The formula for the volume \( V \) of a sphere (which is the shape of a tennis ball) is given by:
\[
V = \frac{4}{3} \pi r^3
\]
Where \( r \) is the radius of the sphere.
1. **Calculate the volume of one tennis ball:**
- Given the radius \( r = 2.5 \) inches, we can substitute this value into the formula. Using \( \pi \approx 3.14 \):
\[
V = \frac{4}{3} \cdot 3.14 \cdot (2.5)^3
\]
First, calculate \( (2.5)^3 \):
\[
(2.5)^3 = 15.625
\]
Now plug this value back into the volume formula:
\[
V = \frac{4}{3} \cdot 3.14 \cdot 15.625
= \frac{4 \cdot 3.14 \cdot 15.625}{3}
\]
Calculate \( 4 \cdot 3.14 \):
\[
4 \cdot 3.14 = 12.56
\]
Now multiply by 15.625:
\[
12.56 \cdot 15.625 \approx 196.09375
\]
Finally, divide by 3:
\[
V \approx \frac{196.09375}{3} \approx 65.36458333
\]
So the volume of one tennis ball is approximately \( 65.36 \) cubic inches.
2. **Calculate the total volume for a dozen (12) tennis balls:**
\[
V_{\text{total}} = 12 \cdot V \approx 12 \cdot 65.36 \approx 784.32 \text{ cubic inches}
\]
3. **Calculate the empty space in the container:**
The volume of the container is given as \( 3200 \) cubic inches. To find the empty space, subtract the total volume of the tennis balls from the volume of the container:
\[
\text{Empty space} = \text{Volume of container} - \text{Volume of tennis balls}
\]
\[
\text{Empty space} = 3200 - 784.32 \approx 2415.68
\]
4. **Round the empty space to the nearest whole number:**
\[
\text{Empty space} \approx 2416 \text{ cubic inches}
\]
Thus, the amount of empty space in the container is approximately **2416 cubic inches**.