Question

If the inside diameter of a cylindrical garden hose is 1 inch, what is the length, in inches, of a straight hose that can hold a maximum of 1 gallon of water? (1 gallon = 231 cubic inches)

• A. \( 231\pi \)
• B. \( \frac{231}{\pi} \)
• C. 924
• D. \( 924\pi \)
• E. \( \frac{924}{\pi} \)

Hint:

Pay close attention to the difference between diameter and radius. This is a common trap. The radius is always half the diameter. Also, when solving equations involving decimals like 0.25 or 0.5, converting them to fractions (\(1/4\) or \(1/2\)) can simplify the algebra.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem requires us to find the length of a cylinder given its volume and diameter. The garden hose is modeled as a cylinder.
Step 2: Key Formula or Approach:
The formula for the volume of a cylinder is \( V = \pi r^2 h \), where \(r\) is the radius and \(h\) is the height (or length in this case). We are given the diameter, so we must first calculate the radius: \( r = \frac{\text{diameter}}{2} \). We are given the volume in gallons, so we must convert it to cubic inches.
Step 3: Detailed Explanation:
1. Identify the given information:

Volume (\(V\)) = 1 gallon = 231 cubic inches.
Inside diameter = 1 inch.
2. Calculate the radius (\(r\)): \[ r = \frac{\text{diameter}}{2} = \frac{1 \text{ inch}}{2} = 0.5 \text{ inches} \] 3. Set up the volume formula: We need to find the length of the hose, which is the height (\(h\)) of the cylinder. \[ V = \pi r^2 h \] 4. Substitute the known values into the formula: \[ 231 = \pi (0.5)^2 h \] \[ 231 = \pi (0.25) h \] 5. Solve for h: To isolate \(h\), divide both sides by \( \pi(0.25) \). \[ h = \frac{231}{0.25\pi} \] It is often easier to work with fractions. \( 0.25 = \frac{1}{4} \). \[ h = \frac{231}{\frac{1}{4}\pi} = \frac{231}{\frac{\pi}{4}} \] To divide by a fraction, we multiply by its reciprocal: \[ h = 231 \times \frac{4}{\pi} = \frac{231 \times 4}{\pi} \] \[ h = \frac{924}{\pi} \] Step 4: Final Answer:
The length of the hose is \( \frac{924}{\pi} \) inches.