Question 5

Explanation

The basic principle used here is Archimedes' principle. When balls are inserted into a cylinder full of water, water spills out. The volume of the spilled water is equal to the total volume of the balls inserted. Thus, the volume of water remaining in the cylinder is the difference between the initial volume of the cylinder and the total volume of the balls.

Volume of the Cylinder (Initial Water Volume):

Volume of One Ball:

Determining the Number of Balls (n):

It is given that the volume of water remaining is $64\pi \text{ cm}^3$. We use the equation:

Dividing both sides by $\pi$ to simplify the calculation:

Since the number of balls $n$ must be an integer and cannot exceed the cylinder's capacity, we take the nearest integer rounded down, which is $17$. If we were to insert $18$ balls, the total volume of the balls would exceed the volume of the cylinder ($18 \times 288\pi = 5184\pi > 5120\pi$), which is impossible.

Thus, the number of balls inserted is $17$ balls.