Answer:
### Volume of the Gym:
The volume of the gym is given by the formula for the volume of a rectangular prism:
\[ \text{Volume}_{\text{Gym}} = \text{Length} \times \text{Width} \times \text{Height} \]
\[ \text{Volume}_{\text{Gym}} = 120 \, \text{ft} \times 80 \, \text{ft} \times 30 \, \text{ft} \]
Calculate the volume of the gym.
### Volume of the Basketball:
The volume of a sphere is given by the formula:
\[ \text{Volume}_{\text{Basketball}} = \frac{4}{3} \pi \left(\frac{\text{Diameter}}{2}\right)^3 \]
Convert the diameter from inches to feet and then calculate the volume of the basketball.
### Packing Space per Basketball:
The packing space per basketball is given as 190% of the volume of a single basketball. Calculate this value.
### Number of Basketballs Needed:
To find the number of basketballs needed to fill the gym, divide the volume of the gym by the packing space per basketball.
Now, let's calculate each part step by step.
### Volume of the Gym:
\[ \text{Volume}_{\text{Gym}} = 120 \, \text{ft} \times 80 \, \text{ft} \times 30 \, \text{ft} = 288,000 \, \text{ft}^3 \]
### Volume of the Basketball:
Given the diameter is 9 inches, convert it to feet: \( \frac{9}{12} \) feet.
\[ \text{Volume}_{\text{Basketball}} = \frac{4}{3} \pi \left(\frac{9}{24}\right)^3 \approx 0.1435 \, \text{ft}^3 \]
### Packing Space per Basketball:
\[ \text{Packing Space per Basketball} = 1.9 \times \text{Volume}_{\text{Basketball}} \approx 0.2727 \, \text{ft}^3 \]
### Number of Basketballs Needed:
\[ \text{Number of Basketballs} = \frac{\text{Volume}_{\text{Gym}}}{\text{Packing Space per Basketball}} \approx 1,056 \]
So, approximately 1,056 basketballs are needed to fill the gym.
