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?
*(Note: The fraction in the image is blurry but is assumed to be 1/2 to match one of the answer choices.)*

• A. \(\frac{16}{9\pi}\)
• B. \(\frac{4}{\sqrt{\pi}}\)
• C. \(\sqrt{\frac{12}{\pi}}\)
• D. \(\sqrt{\frac{2}{\pi}}\)
• E. \(4\sqrt{\frac{2}{\pi}}\)

Hint:

Be careful to distinguish between radius and diameter. Many geometry problems will ask for the diameter after you've calculated the radius, so always double-check what the question is asking for before selecting an answer.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The problem relates the partial volume of a cylinder to its total volume and dimensions (height, radius). We need to find the total volume first, then use the volume formula to solve for the radius, and finally find the diameter.
Step 2: Key Formula or Approach:
The volume of a cylinder is given by \(V = \pi r^2 h\), where r is the radius and h is the height.
Step 3: Detailed Explanation:
We are given that the container is \(\frac{1}{2}\) full and the volume of the water is 36 cubic inches. Let \(V_{total}\) be the total volume of the cylinder. \[ \frac{1}{2} V_{total} = 36 \] First, find the total volume of the cylinder: \[ V_{total} = 36 \times 2 = 72 \text{ cubic inches} \] Now use the cylinder volume formula with the total volume and the given height \(h=9\) inches. \[ V_{total} = \pi r^2 h \] \[ 72 = \pi r^2 (9) \] To solve for r, first divide both sides by 9: \[ 8 = \pi r^2 \] Isolate \(r^2\): \[ r^2 = \frac{8}{\pi} \] Take the square root of both sides to find the radius r: \[ r = \sqrt{\frac{8}{\pi}} = \sqrt{\frac{4 \times 2}{\pi}} = 2\sqrt{\frac{2}{\pi}} \] The question asks for the diameter, which is twice the radius (\(d = 2r\)). \[ d = 2 \times \left(2\sqrt{\frac{2}{\pi}}\right) = 4\sqrt{\frac{2}{\pi}} \] Step 4: Final Answer:
The diameter of the base of the cylinder is \(4\sqrt{\frac{2}{\pi}}\) inches.