Question

A container in the shape of a right circular cylinder is \( \frac{1}{2} \) full of water. If the volume of water in the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of the base of the cylinder, in inches?

• A. \( \frac{16}{9\pi} \)
• B. \( \frac{4}{\sqrt{\pi}} \)
• C. \( \frac{36}{\sqrt{2\pi}} \)
• D. \( \frac{36}{\sqrt{\pi}} \)
• E. \( \frac{4}{\sqrt{\pi}} \)

Hint:

For problems involving volumes of cylinders, remember to rearrange the volume formula to solve for unknowns.

Answer 1

The Correct Option is B

Step 1: Volume of a cylinder.
The volume of a cylinder is given by the formula: \[ V = \pi r^2 h \] Where:
- \( V \) is the volume,
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.
Step 2: Use the known values.
We are given that the cylinder is \( \frac{1}{2} \) full of water, so: \[ \frac{1}{2} \times \pi r^2 \times 9 = 36 \] Simplifying: \[ \pi r^2 \times 9 = 72 \quad \implies \quad r^2 = \frac{72}{9\pi} = \frac{8}{\pi} \] Step 3: Calculate the diameter.
The diameter is \( 2r \), so: \[ r = \sqrt{\frac{8}{\pi}} = \frac{4}{\sqrt{\pi}} \] Thus, the diameter is: \[ 2r = 2 \times \frac{4}{\sqrt{\pi}} = \frac{8}{\sqrt{\pi}} \] \[ \boxed{\frac{4}{\sqrt{\pi}}} \]