Final answer:
Approximately 937 four-inch diameter balls can fit in a toy machine shaped like a rectangular prism when it is halfway full.
Step-by-step explanation:
The question asks us to calculate the number of 4-inch diameter balls that can fit in a toy machine shaped like a rectangular prism when it is halfway full. The machine's dimensions are 32 inches in length, 32 inches in width, and 46 inches in height. Since the machine is only halfway full, we will only consider the volume up to 23 inches high.
First, calculate the volume of the machine that holds the balls when it is halfway filled:
- Volume of the machine's half = length × width × (half of the height)
- Volume of the machine's half = 32 in × 32 in × 23 in
- Volume of the machine's half = 23,552 cubic inches
Next, calculate the volume of one ball:
- Volume of a sphere = ⅔ π × radius^3
- Since the diameter is 4 inches, the radius is 2 inches.
- Volume of one ball = ⅔ π × 2^3 = π × 8 cubic inches ≈ 25.13 cubic inches
Finally, divide the half volume of the machine by the volume of one ball to find out how many balls can fit:
- Number of balls = Volume of machine's half / Volume of one ball
- Number of balls ≈ 23,552 / 25.13
- Number of balls ≈ 937 balls
Therefore, approximately 937 balls can fit in the machine when it is halfway full.