The volume of a container measures 3,200 in.3 and contains a dozen tennis balls. Each tennis ball has a radius of 2.5 in. How much space in the container is empty? Use 3.14 as an approximation for pi. Round to the nearest whole number, if necessary.(1 point) ____in.3

Answer 1

To find the empty space in the container, we will first calculate the volume of one tennis ball and then the total volume of the dozen (12) tennis balls. Finally, we will subtract the total volume of the tennis balls from the volume of the container.

1. Calculate the volume of one tennis ball: The formula for the volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.

   The radius \( r \) of each tennis ball is 2.5 inches. Using \( \pi \approx 3.14 \): \[ V = \frac{4}{3} \times 3.14 \times (2.5)^3 \] First, calculate \( (2.5)^3 \): \[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]

   Now, substitute this value back into the volume formula: \[ V = \frac{4}{3} \times 3.14 \times 15.625 \] Calculate \( \frac{4}{3} \times 3.14 \): \[ \frac{4 \times 3.14}{3} \approx \frac{12.56}{3} \approx 4.18667 \] Now multiply this by 15.625: \[ V \approx 4.18667 \times 15.625 \approx 65.282 \]

   So the volume of one tennis ball is approximately: \[ V \approx 65.28 \text{ in}^3 \]

2. Calculate the volume of a dozen tennis balls: \[ \text{Total Volume} = 12 \times 65.28 \approx 783.36 \text{ in}^3 \]

3. Calculate the empty space in the container: The total volume of the container is 3200 in³. The empty space is given by: \[ \text{Empty Space} = \text{Volume of Container} - \text{Total Volume of Tennis Balls} \] \[ \text{Empty Space} = 3200 - 783.36 \approx 2416.64 \text{ in}^3 \]

Rounding to the nearest whole number, the empty space in the container is: \[ \boxed{2417} \text{ in}^3 \]

Answer 2

that's wrong

Answer 3

I apologize for the oversight. Let's go through the calculations again step by step.

1. Calculate the volume of one tennis ball using the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \] where the radius \( r \) is 2.5 inches and we will use \( \pi \approx 3.14 \).

   First, calculate \( (2.5)^3 \): \[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \]

   Then plug this value into the volume formula: \[ V = \frac{4}{3} \times 3.14 \times 15.625 \]

   Calculate \( \frac{4}{3} \times 3.14 \): \[ \frac{4 \times 3.14}{3} = \frac{12.56}{3} \approx 4.18667 \]

   Now, calculate the volume: \[ V \approx 4.18667 \times 15.625 \approx 65.28875 \] So the volume of one tennis ball is approximately: \[ V \approx 65.29 \text{ in}^3 \]

2. Calculate the volume of a dozen tennis balls: \[ \text{Total Volume of 12 Tennis Balls} = 12 \times 65.29 \approx 783.48 \text{ in}^3 \]

3. Calculate the empty space in the container: The total volume of the container is 3200 in³. Therefore, the empty space is: \[ \text{Empty Space} = \text{Volume of Container} - \text{Total Volume of Tennis Balls} \] \[ \text{Empty Space} = 3200 - 783.48 \approx 2416.52 \text{ in}^3 \]

Rounding to the nearest whole number, the empty space in the container is: \[ \boxed{2417} \text{ in}^3 \]

If you believe there is an error regarding values or a specific part of the calculation, please let me know so I can correct it.