Math / Geometry

QuestionA 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 $\pi r^{2} h$ $\frac{V}{36 \pi}$ $\frac{V}{6 \pi}$ $\frac{V}{8 \pi}$ $\frac{V}{48 \pi}$ $\frac{V}{27 \pi}$

Studdy Solution

STEP 1

1. The tank is a right circular cylinder.
2. The tank is filled with water to 75% of its total volume $V$ .
3. The radius of the tank is $6$ inches.
4. The height of the tank is $h$ inches.
5. The volume of a cylinder is given by the formula $V = \pi r^2 h$ .

STEP 2

1. Express the total volume of the tank in terms of its dimensions.
2. Determine the volume of water in the tank.
3. Solve for the height of the water in the tank.

STEP 3

Express the total volume of the tank in terms of its dimensions:
The volume of the cylinder is given by:
$V = \pi r^2 h$
Substitute the radius $r = 6$ inches:
$V = \pi (6)^2 h$ $V = 36\pi h$

STEP 4

Determine the volume of water in the tank:
The tank is filled to 75% of its total volume:
$\text{Volume of water} = 0.75V$

STEP 5

Solve for the height of the water in the tank:
Let $h_w$ be the height of the water. The volume of the water can also be expressed as:
$\text{Volume of water} = \pi (6)^2 h_w$ $= 36\pi h_w$
Set the expressions for the volume of water equal:
$0.75V = 36\pi h_w$
Substitute $V = 36\pi h$ :
$0.75(36\pi h) = 36\pi h_w$
Simplify the equation:
$27\pi h = 36\pi h_w$
Divide both sides by $36\pi$ :
$h_w = \frac{27\pi h}{36\pi}$ $h_w = \frac{27}{36} h$ $h_w = \frac{3}{4} h$
Since $h_w = \frac{3}{4} h$ , we can express $h_w$ in terms of $V$ :
$h_w = \frac{3}{4} \left(\frac{V}{36\pi}\right)$ $h_w = \frac{3V}{144\pi}$ $h_w = \frac{V}{48\pi}$
Thus, the height of the water in the tank is represented by:
$\boxed{\frac{V}{48\pi}}$

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