$1413\pi$
The toy consists of two parts: a solid cone and a frustum. The total surface area of the toy is the sum of the curved surface area of the cone, the curved surface area of the frustum, and the area of the larger base of the frustum.
1. Calculate Cone Details
Given:
- Cone radius (\(r_1\)) = 5 cm
- Cone height (\(h_1\)) = 12 cm
Calculate the slant height (\(l_1\)) of the cone using the Pythagorean theorem:
\(l_1 = \sqrt{r_1^2 + h_1^2}\)
\(l_1 = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}\)
Calculate the Curved Surface Area (CSA) of the cone:
\(CSA_{\text{cone}} = \pi r_1 l_1\)
\(CSA_{\text{cone}} = \pi \times 5 \times 13 = 65\pi \text{ cm}^2\)
2. Calculate Frustum Details
Given:
- Frustum smaller base radius (\(r_1\)) = 5 cm
- Frustum larger base radius (\(r_2\)) = 22 cm
- Frustum slant height (\(l_2\)) = 32 cm
Calculate the Curved Surface Area (CSA) of the frustum:
\(CSA_{\text{frustum}} = \pi (r_1 + r_2) l_2\)
\(CSA_{\text{frustum}} = \pi (5 + 22) \times 32 = \pi \times 27 \times 32 = 864\pi \text{ cm}^2\)
Calculate the Area of the larger base of the frustum:
\(Area_{\text{base}} = \pi r_2^2\)
\(Area_{\text{base}} = \pi \times 22^2 = 484\pi \text{ cm}^2\)
3. Calculate Total Surface Area
The total surface area (TSA) of the toy is:
\(TSA = CSA_{\text{cone}} + CSA_{\text{frustum}} + Area_{\text{base}}\)
\(TSA = 65\pi + 864\pi + 484\pi\)
\(TSA = (65 + 864 + 484)\pi\)
\(TSA = 1413\pi \text{ cm}^2\)
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