A cylinder has volume V, height h, and base diameter d. Which of the following represents d in terms of V and h?

Options

A. `d=sqrt(Vpih)`
B. `d=sqrt((v)/(pih))`
C. `d=sqrt((2V)/(pih))`
D. `d=sqrt((4V)/(pih))`

Answer

To find the diameter \( d \) of a cylinder in terms of its volume \( V \) and height \( h \), we can follow these steps:

### Step 1: Write the formula for the volume of a cylinder.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base of the cylinder and \( h \) is the height.

### Step 2: Relate the radius to the diameter.
The radius \( r \) is half of the diameter \( d \):
\[ r = \frac{d}{2} \]

### Step 3: Substitute the expression for \( r \) into the volume formula.
Substituting \( r = \frac{d}{2} \) into the volume formula, we get:
\[ V = \pi \left(\frac{d}{2}\right)^2 h \]

### Step 4: Simplify the equation.
This simplifies to:
\[ V = \pi \left(\frac{d^2}{4}\right) h \]
which can be rewritten as:
\[ V = \frac{\pi d^2 h}{4} \]

### Step 5: Solve for \( d^2 \).
To isolate \( d^2 \), we multiply both sides by 4:
\[ 4V = \pi d^2 h \]
Now, divide both sides by \( \pi h \):
\[ d^2 = \frac{4V}{\pi h} \]

### Step 6: Take the square root to find \( d \).
Taking the square root of both sides, we find:
\[ d = \sqrt{\frac{4V}{\pi h}} \]

### Final Result:
Thus, the diameter \( d \) in terms of the volume \( V \) and height \( h \) is:
\[ d = \sqrt{\frac{4V}{\pi h}} \]