lincoln is filling a cylindrical shaped box woth sphere shaped balls. the box has a radius of 5 inches and a height of 10 inches. each ball has a radius of 2 inches. how many balls can fit into the box? round down to the nearest whole number.
To determine how many balls can fit into the box, we first need to calculate the volume of the box and the volume of a single ball.
The volume of the box can be calculated using the formula for the volume of a cylinder:
V_box = π * r^2 * h
Where:
- V_box is the volume of the box,
- π is a mathematical constant, approximately equal to 3.14159,
- r is the radius of the box (5 inches), and
- h is the height of the box (10 inches).
V_box = 3.14159 * 5^2 * 10
V_box = 3.14159 * 25 * 10
V_box = 3.14159 * 250
V_box ≈ 785.39875 cubic inches
Next, we calculate the volume of a single ball using the formula for the volume of a sphere:
V_ball = (4/3) * π * r^3
Where:
- V_ball is the volume of a single ball, and
- r is the radius of the ball (2 inches).
V_ball = (4/3) * 3.14159 * 2^3
V_ball = (4/3) * 3.14159 * 8
V_ball = (4/3) * 25.13272
V_ball = 33.510321 cubic inches
Finally, we divide the volume of the box by the volume of a single ball to find the number of balls that can fit:
Number of balls = V_box / V_ball
Number of balls ≈ 785.39875 / 33.510321
Number of balls ≈ 23.446
Since we need to round down to the nearest whole number, we can conclude that approximately 23 balls can fit into the box.