The problem involves a cylindrical water tank filled to 75% of its total volume. We need to determine the height of the water level based on the provided volume formula for a cylinder.

Key Information:

• The total volume $V$ of a cylinder is given by: $V = \pi r^2 h$ where $r$ is the radius of the cylinder and $h$ is its height.

• The radius $r$ is 6 inches.

• We are asked to find the height of the water when the tank is filled to 75% of its volume.

Step-by-Step Solution:

1. Total Volume of the Tank: Using the volume formula for a cylinder, the total volume $V_{\text{total}}$ is: $V_{\text{total}} = \pi r^2 h = \pi (6)^2 h = 36\pi h$

2. Volume of Water (75% of Total Volume): Since the tank is filled to 75% of its volume, the volume of water $V_{\text{water}}$ is: $V_{\text{water}} = 0.75 \times V_{\text{total}} = 0.75 \times 36\pi h = 27\pi h$

3. Formula for the Height of the Water: The height of the water level can be found by setting up the volume formula for the water, where the water's volume is also a cylinder but with height $h_{\text{water}}$ instead of $h$. So: $V_{\text{water}} = \pi r^2 h_{\text{water}}$ Substituting the radius $r = 6$: $V_{\text{water}} = \pi (6)^2 h_{\text{water}} = 36\pi h_{\text{water}}$

4. Solving for $h_{\text{water}}$: Set the two expressions for the water's volume equal to each other: $27\pi h = 36\pi h_{\text{water}}$ Simplifying: $h_{\text{water}} = \frac{27\pi h}{36\pi} = \frac{3h}{4}$

Thus, the height of the water level is $\frac{3h}{4}$, where $h$ is the total height of the tank.

Checking the Options:

The correct answer should match the expression for $h_{\text{water}}$. Since the height is $\frac{3h}{4}$, we can conclude that the correct answer is not directly listed in terms of $h$, but it relates to this.

Do you need more details, or should I clarify something further?

Here are some questions you might want to explore based on this problem:

1. How would the height of the water change if the tank were filled to 50% instead of 75%?
2. What would be the height of the water if the radius were doubled?
3. How does the volume of a cylinder change if the height is halved but the radius is doubled?
4. How can the formula for the volume of a cylinder be derived geometrically?
5. What is the impact of increasing both the radius and height on the volume?

Tip: When solving volume-related problems for cylindrical shapes, always make sure to account for the correct proportional relationships between radius, height, and volume!