To determine the relationship between the volumes of the blue and red exercise balls, we can use the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
where \(r\) is the radius of the sphere.
First, we need to find the radius of each ball. The radius is half of the diameter.
For the blue ball:
- Diameter = 56 cm
- Radius = \( \frac{56}{2} = 28 \) cm
For the red ball:
- Diameter = 76 cm
- Radius = \( \frac{76}{2} = 38 \) cm
Now we can calculate the volume of each ball.
Volume of the blue ball: \[ V_{blue} = \frac{4}{3} \pi (28)^3 \]
Calculating \(28^3\): \[ 28 \times 28 \times 28 = 21952 \]
So, \[ V_{blue} \approx \frac{4}{3} \pi (21952) \approx 92019.5 \pi \]
Volume of the red ball: \[ V_{red} = \frac{4}{3} \pi (38)^3 \]
Calculating \(38^3\): \[ 38 \times 38 \times 38 = 54872 \]
So, \[ V_{red} \approx \frac{4}{3} \pi (54872) \approx 230312.5 \pi \]
Now, we can find the ratio of the volumes: \[ \frac{V_{red}}{V_{blue}} = \frac{230312.5 \pi}{92019.5 \pi} = \frac{230312.5}{92019.5} \]
Calculating this gives approximately: \[ \frac{230312.5}{92019.5} \approx 2.5 \]
This indicates that the volume of the red ball is about 2.5 times larger than the volume of the blue ball.
Therefore, the correct statement is: The volume of the red ball is about 2.5 times larger than the volume of the blue ball.