The height of sand in a cylinder-shaped can drops 3 inches foot of sand is poured out. What is the diameter, in inches of the cylinder?

Options

A. `(2)/(sqrt(pi))`
B. `(4)/(sqrt(pi))`
C. `(16)/(pi)`
D. `(48)/(sqrt(pi))`

Answer

To solve the problem, we need to find the diameter of a cylinder-shaped can based on the information given about the height drop when a certain volume of sand is poured out. Here’s a step-by-step solution:

### Step 1: Understand the relationship between volume and height
When 1 cubic foot of sand is poured out, it causes the height of the sand in the cylinder to drop by 3 inches. The volume of the cylinder can be expressed as:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 2: Convert measurements to consistent units
Since the height drop is given in inches, we need to convert it to feet for consistency. We know:
- 1 inch = 1/12 feet
- Therefore, 3 inches = \( \frac{3}{12} = 0.25 \) feet.

### Step 3: Set up the volume equation
The volume of sand that was poured out is 1 cubic foot. This volume corresponds to the volume of the cylinder that has been emptied, which can be expressed as:
\[ V = \pi r^2 h \]
Substituting \( h = 0.25 \) feet:
\[ 1 = \pi r^2 (0.25) \]

### Step 4: Solve for \( r^2 \)
Rearranging the equation gives:
\[ r^2 = \frac{1}{\pi \cdot 0.25} \]
\[ r^2 = \frac{1}{0.25\pi} = \frac{4}{\pi} \]

### Step 5: Calculate the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{\frac{4}{\pi}} = \frac{2}{\sqrt{\pi}} \]

### Step 6: Convert radius to inches
Since we need the diameter in inches, we first convert the radius from feet to inches. We know:
- 1 foot = 12 inches
So, the radius in inches is:
\[ r = \frac{2}{\sqrt{\pi}} \times 12 = \frac{24}{\sqrt{\pi}} \]

### Step 7: Calculate the diameter
The diameter \( d \) is twice the radius:
\[ d = 2r = 2 \times \frac{24}{\sqrt{\pi}} = \frac{48}{\sqrt{\pi}} \]

### Final Answer
The diameter of the cylinder is:
\[ d = \frac{48}{\sqrt{\pi}} \text{ inches} \]

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