A circular cylindrical water tank is filled with water to 75 percent of its total volume of $V$ cubic inches. The radius of the tank is 6 inches, and the height of the tank is $h$ inches. Which of the following represents the height, in inches, of the water in the tank? (Note: The volume of a cylinder with radius $r$ and height $h$ is given by $\pi r^2 h$ .)

Question :

A circular cylindrical water tank is filled with water to 75 percent of its total volume of $v$ cubic inches. the radius of the tank is 6 inches, and the height of the tank is $h$ inches. which of the following represents the height, in inches, of the water in the tank? (note: the volume of a cylinder with radius $r$ and height $h$ is given by $\pi r^2 h$ .)

A circular cylindrical water tank is filled with water to 75 percent of its tota | Doubtlet.com

Solution:

Neetesh Kumar | October 23, 2024

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CLEPS College Algebra Guide Question 78
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Step-by-step solution:

We are given the total volume of the cylinder ( V ), and we know the tank is filled to 75% of its total volume. The formula for the volume of a cylinder is:

$V = \pi r^2 h$

We need to find the height of the water, which is 75% of the total volume.

Step 1: Write the equation for the volume of the water

The volume of the water is 75% of the total volume, so the volume of the water is:

$\text{Volume of water} = 0.75 \times V$

Using the volume formula for a cylinder, the volume of the water is also given by:

$0.75 \times V = \pi r^2 h_{\text{water}}$

where ( h_{\text{water}} ) is the height of the water. We are given the radius ( r = 6 ) inches.

Step 2: Substitute the radius into the equation

Substitute ( r = 6 ) into the equation:

$0.75 \times V = \pi (6)^2 h_{\text{water}}$

Simplify:

$0.75 \times V = \pi \times 36 \times h_{\text{water}}$

Step 3: Solve for $h_{\text{water}}$

To find the height of the water, divide both sides by ( 36\pi ):

$h_{\text{water}} = \frac{0.75 \times V}{36\pi}$

Simplify the right-hand side:

$h_{\text{water}} = \frac{3V}{144\pi} = \frac{V}{48\pi}$

Final Answer:

The height of the water is:

$\text{(E) } \frac{V}{48\pi}$