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Answer :
Final answer:
Approximately 6,814 basketballs, each with a 24-inch diameter, are required to fill a gym that measures 78 feet by 42 feet by 30 feet.
Explanation:
Calculating the Number of Basketballs to Fill a Gym
The question asks us to calculate how many basketballs, each with a diameter of 24 inches, are needed to fill a gym that is 78 feet long, 42 feet wide, and 30 feet high. To find the number of basketballs, we need to determine the volume of the gym and the volume of a single basketball and then divide the gym's volume by the basketball's volume.
Firstly, convert the gym's dimensions to inches since the basketball's diameter was given in inches (1 foot = 12 inches):
- Gym length in inches: 78 feet × 12 inches/foot = 936 inches
- Gym width in inches: 42 feet × 12 inches/foot = 504 inches
- Gym height in inches: 30 feet × 12 inches/foot = 360 inches
Next, calculate the gym's volume in cubic inches:
- Gym volume: 936 inches × 504 inches × 360 inches = 169,574,400 cubic inches
The volume of a basketball is given by the formula for the volume of a sphere, V = (4/3)πr³, where r is the radius of the sphere. The radius of the basketball is half of its diameter, so r = 24 inches / 2 = 12 inches. Now calculate the volume of the basketball:
- Basketball volume: (4/3)π(12 inches)³ ≈ (4/3)π(1,728 cubic inches) ≈ 24,883.3 cubic inches (rounded to one decimal place)
To find the number of basketballs needed to fill the gym, divide the gym's volume by the volume of a single basketball:
- Number of basketballs: 169,574,400 cubic inches / 24,883.3 cubic inches ≈ 6,814
Therefore, approximately 6,814 basketballs would be needed to fill the gym.
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