A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the height of the cylindrical part to the height of the conical part is :

Step 1: Define variables and formulas

Let $r$ be the common radius of the cylindrical and conical parts. Let $h_c$ be the height of the cylindrical part and $h_{co}$ be the height of the conical part. We write down the formulas for the volume of a cylinder ($V_c$) and the volume of a cone ($V_{co}$).

Step 2: Equate the volumes

The problem states that the volume of the cylindrical part is equal to the volume of the conical part. We set the two volume formulas equal to each other.

Step 3: Simplify the equation

We can cancel out the common terms $\pi r^2$ from both sides of the equation, as $r$ cannot be zero. This simplifies the relationship between the heights.

Step 4: Find the ratio of heights

To find the ratio of the height of the cylindrical part to the height of the conical part, we rearrange the simplified equation to express $h_c$ divided by $h_{co}$.

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