We can simplify the equation by canceling out common terms like π and r from both sides. This gives us a relationship between the radius and the slant height of the cone.

A solid is hemispherical at the bottom and conical above (of same radius). If the curved surface areas of the two parts are equal, then the ratio of its radius and the height of the conical part is

Answer: The ratio of its radius and the height of the conical part is $1:\sqrt{3}$ .

Step-by-step solution

Step 1: Define variables and formulas

Let $r$ be the common radius of the hemispherical and conical parts. Let $h$ be the height of the conical part and $l$ be its slant height. We recall the formulas for the curved surface area (CSA) of a hemisphere and a cone.

Step 2: Equate curved surface areas

According to the problem statement, the curved surface areas of the hemispherical and conical parts are equal. We set the two CSA formulas equal to each other.

Step 3: Simplify the equation to find slant height

We can simplify the equation by canceling out common terms like $\pi$ and $r$ from both sides. This gives us a relationship between the radius and the slant height of the cone.

Step 4: Relate slant height, radius, and height of cone

For a cone, the slant height, radius, and height form a right-angled triangle. Therefore, we can use the Pythagorean theorem to relate them.

Step 5: Substitute and solve for the ratio

Substitute the value of $l$ from the previous step into the Pythagorean theorem. Then, rearrange the equation to find the ratio of $r$ to $h$ .

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