The volume of a cylinder and cone are in the ratio 3:1. Find their diameters and then compare them when their heights are equal.

Options

A. Diameter of cylinder - 2 times of diameter of cone
B. Diameter of cylinder = Diameter of cone
C. Diameter of cylinder `gt` Diameter of cone
D. Diameter of cylinder `lt` Diameter of cone

Answer

To solve the problem, we need to find the diameters of a cylinder and a cone when their volumes are in the ratio of 3:1 and their heights are equal.

### Step 1: Understand the formulas for the volumes
The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

### Step 2: Set up the ratio of volumes
According to the problem, the ratio of the volumes of the cylinder and the cone is given as:
\[ \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{3}{1} \]
Substituting the volume formulas into the ratio gives:
\[ \frac{\pi r_1^2 h}{\frac{1}{3} \pi r_2^2 h} = \frac{3}{1} \]

### Step 3: Simplify the equation
Since the heights \( h \) of both the cylinder and cone are equal, we can cancel \( h \) and \( \pi \) from both sides:
\[ \frac{r_1^2}{\frac{1}{3} r_2^2} = 3 \]
This simplifies to:
\[ \frac{r_1^2}{r_2^2} = 3 \cdot \frac{1}{3} = 3 \]

### Step 4: Cross-multiply to find the relationship between the radii
Cross-multiplying gives:
\[ r_1^2 = 3 r_2^2 \]

### Step 5: Take the square root to find the radii
Taking the square root of both sides gives:
\[ r_1 = \sqrt{3} r_2 \]

### Step 6: Find the diameters
The diameter \( d \) is twice the radius, so:
\[ d_1 = 2r_1 = 2(\sqrt{3} r_2) = 2\sqrt{3} r_2 \]
\[ d_2 = 2r_2 \]

### Step 7: Compare the diameters
Now, we can compare the diameters:
\[ \frac{d_1}{d_2} = \frac{2\sqrt{3} r_2}{2 r_2} = \sqrt{3} \]
This means that the diameter of the cylinder is \( \sqrt{3} \) times the diameter of the cone.

### Conclusion
Thus, the diameters of the cylinder and cone when their heights are equal are in the ratio of \( \sqrt{3} : 1 \).