Question

A toy is in the form of a cone of radius \( r \) and lateral height \( l \) mounted on a hemisphere of the same radius, and the total height of the toy is \( h \), then the total surface area of the toy is:

• A. \( \pi r (2r + l) \)
• B. \( 2\pi r + l \)
• C. \( \pi r^2 l \)
• D. \( \pi r^2 h \)

Hint:

The total surface area of a cone mounted on a hemisphere is the sum of the surface areas of the cone and the hemisphere, but without counting the base of the cone twice.

Answer 1

The Correct Option is A

Step 1: Understanding the Surface Area of the Toy.
The toy consists of a cone mounted on a hemisphere, both with radius \( r \). The total surface area of the cone is the area of the conical surface, given by: \[ \text{Surface Area of the Cone} = \pi r l \] where \( l \) is the slant height (lateral height) of the cone. The total surface area of the hemisphere is the area of its curved surface, given by: \[ \text{Surface Area of the Hemisphere} = 2 \pi r^2 \] Step 2: Total Surface Area of the Toy.
The total surface area of the toy is the sum of the surface area of the cone and the surface area of the hemisphere. However, the base of the cone is already part of the hemisphere, so we do not count it twice. Therefore, the total surface area is: \[ \text{Total Surface Area} = \text{Surface Area of the Cone} + \text{Surface Area of the Hemisphere} \] \[ \text{Total Surface Area} = \pi r l + 2 \pi r^2 \] \[ \text{Total Surface Area} = \pi r (l + 2r) \] Step 3: Conclusion.
Thus, the total surface area of the toy is \( \pi r (2r + l) \).