A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to tis total height `h ,` when `h=9c mdot`

Explanation

To solve the problem step by step, we will find the rate of change of the volume of a balloon in the shape of a right circular cone surmounted by a hemisphere, with respect to its total height \( h \).

### Step 1: Understand the Geometry
The balloon consists of a cone and a hemisphere. The diameter of the hemisphere is equal to the height of the cone. Let:
- \( r \) = radius of the hemisphere (and the base of the cone)
- \( h_c \) = height of the cone
- \( h_h \) = height of the hemisphere

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