To determine the height of the cylinder that is immersed in the hemisphere, we need to consider the following steps:

Given:

1. Hemisphere:

   • Radius $r = 10 \, \text{cm}$.
   • Volume of the hemisphere:
     $V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (10)^3 = \frac{2000}{3} \pi \, \text{cm}^3.$

2. Cylinder:

   • Radius $R = 8 \, \text{cm}$.
   • Height $h$ (to be determined).
   • Volume of the cylinder: $V_{\text{cylinder}} = \pi R^2 h = \pi (8)^2 h = 64 \pi h \, \text{cm}^3.$

3. The volume of water displaced is equal to the volume of the submerged part of the cylinder. The problem states that the volume of water displaced must be less than the hemisphere's volume, but greater than 900 cm³.

Steps:

1. Determine the displaced water volume: $V_{\text{displaced}} = V_{\text{cylinder}} = 64 \pi h.$ The volume of the displaced water should satisfy the inequality: $900 \, \text{cm}^3 < 64 \pi h < \frac{2000}{3} \pi.$

2. Solve the inequality for $h$:

   • Convert $900 \, \text{cm}^3$ into terms of $\pi$: $900 = 900 / \pi \approx 286.48 \, \text{cm}^3 \, (\text{in terms of volume}).$

Let's clarify the solution to find the height $h$ of the immersed cylinder:

Steps to Find Height $h$:

1. The displaced water volume equals the cylinder volume: $V_{\text{displaced}} = V_{\text{cylinder}} = 64 \pi h.$

2. Apply the volume inequality: $900 \, \text{cm}^3 < 64 \pi h < \frac{2000}{3} \pi.$

3. Express the limits for $h$:

   • Divide through by $64 \pi$: $\frac{900}{64 \pi} < h < \frac{\frac{2000}{3} \pi}{64 \pi}.$

4. Simplify each side:

   • For the left side: $h > \frac{900}{64 \pi} = \frac{900}{64 \cdot 3.1416} \approx \frac{900}{201.06} \approx 4.48 \, \text{cm}.$

   • For the right side: $h < \frac{\frac{2000}{3} \pi}{64 \pi} = \frac{2000}{3 \cdot 64} = \frac{2000}{192} \approx 10.42 \, \text{cm}.$

5. Final result for $h$: $4.48 \, \text{cm} < h < 10.42 \, \text{cm}.$

Explanation:

The height $h$ of the cylinder immersed in the hemisphere lies between 4.48 cm and 10.42 cm, satisfying the conditions of the problem where the displaced water volume is between 900 cm³ and the total volume of the hemisphere.

Would you like further clarification or additional calculations? Let me know!

Follow-Up Questions:

1. What is the formula for the volume of a hemisphere?
2. How is the volume of a cylinder derived?
3. Why is the water displacement equal to the submerged volume of the cylinder?
4. What happens if the cylinder height exceeds the hemisphere's radius?
5. Can this problem be solved using graphical methods?

Tip:

Always double-check units and ensure volumes are consistently compared in terms of $\pi$ when dealing with curved geometries.