A cylindrical glass 7 cm in diameter and 10 cm tall is filled with water to a height of 9 cm.
If a ball 5 cm in diameter is dropped into the class and sinks to the bottom, will the water in the glass overflow?
To find out if the water will overflow, we need to calculate the volume of the water in the glass before and after the ball is dropped in.
The initial volume of water in the glass can be calculated using the formula for the volume of a cylinder:
V_initial = πr^2h
V_initial = π(3.5)^2(9)
V_initial ≈ 346.36 cm^3
Next, we need to calculate the volume of the ball:
V_ball = (4/3)πr^3
V_ball = (4/3)π(2.5)^3
V_ball ≈ 65.45 cm^3
The total volume of water and ball in the glass after the ball is dropped in can be calculated as:
V_total = V_initial + V_ball
V_total = 346.36 + 65.45
V_total ≈ 411.81 cm^3
The volume of the glass is given by the formula for the volume of a cylinder:
V_glass = πr^2h
V_glass = π(3.5)^2(10)
V_glass ≈ 384.85 cm^3
Since the total volume of water and ball in the glass (411.81 cm^3) is greater than the volume of the glass itself (384.85 cm^3), the water in the glass will overflow.