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 :
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Step-by-Step Solution
Step 1: Define variables and formulas
Let r be the common radius of the cylindrical and conical parts. Let hc be the height of the cylindrical part and hco be the height of the conical part. We write down the formulas for the volume of a cylinder (Vc) and the volume of a cone (Vco).
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 πr2 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 hc divided by hco.