A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. find the total surface area and the volume of the toy.

To solve the problem of finding the total surface area and the volume of the toy, which consists of a cone mounted on a hemisphere, we need to break the problem into parts: the hemisphere and the cone.

Given:
Radius of the hemisphere (r) = 3.5 cm
Total height of the toy = 15.5 cm
The height of the cone can be calculated by subtracting the radius of the hemisphere from the total height.
Step 1: Height of the cone
The height of the cone is: Height of the cone = Total height - Radius of the hemisphere Height of the cone = 15.5 cm - 3.5 cm = 12 cm

Step 2: Volume of the toy
The toy consists of two parts: a hemisphere and a cone.

Volume of the hemisphere:
The formula for the volume of a hemisphere is: Volume of hemisphere = (2/3) * π * r³ Volume of hemisphere = (2/3) * π * (3.5)³ Volume of hemisphere = (2/3) * π * 42.875 Volume of hemisphere ≈ 89.75 cm³

Volume of the cone:
The formula for the volume of a cone is: Volume of cone = (1/3) * π * r² * h Volume of cone = (1/3) * π * (3.5)² * 12 Volume of cone = (1/3) * π * 12.25 * 12 Volume of cone ≈ 462.21 cm³

Total volume of the toy:
Total volume = Volume of hemisphere + Volume of cone Total volume = 89.75 cm³ + 462.21 cm³ Total volume ≈ 551.96 cm³

Step 3: Surface area of the toy
The total surface area consists of the curved surface area of the hemisphere and the curved surface area of the cone.

Surface area of the hemisphere:
The formula for the surface area of a hemisphere (excluding the base) is: Surface area of hemisphere = 2 * π * r² Surface area of hemisphere = 2 * π * (3.5)² Surface area of hemisphere = 2 * π * 12.25 Surface area of hemisphere ≈ 76.96 cm²

Surface area of the cone:
The formula for the lateral (curved) surface area of a cone is: Surface area of cone = π * r * l Where l is the slant height of the cone. To find l, we use the Pythagorean theorem: l = √(r² + h²) l = √((3.5)² + (12)²) l = √(12.25 + 144) l = √156.25 l ≈ 12.5 cm

Surface area of cone = π * 3.5 * 12.5 Surface area of cone ≈ 137.44 cm²

Total surface area of the toy:
The total surface area is the sum of the curved surface area of the hemisphere and the cone. Note that the flat circular base of the hemisphere is not part of the external surface area because it is covered by the cone.

Total surface area = Surface area of hemisphere + Surface area of cone Total surface area ≈ 76.96 cm² + 137.44 cm² Total surface area ≈ 214.4 cm²

Final Answers:
Total volume of the toy ≈ 551.96 cm³
Total surface area of the toy ≈ 214.4 cm²