Question 3 of 13intermediate🔧 ApplyNumerical3 marks

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

Correct Answer

214.5 cm²

Exercise: EXERCISE 12.1 | Q: 3 | (Page Page 6)
For More Understanding

Explanation

This question is from Exercise 12.1, Question 3. The toy consists of a cone mounted on a hemisphere. The key insight from the textbook is that the total surface area of such a toy equals the curved surface area of the hemisphere plus the curved surface area of the cone. The flat circular base where they join is not part of the external surface. Students must find the slant height of the cone using the Pythagorean theorem before calculating the CSA of the cone.

Solution Steps

  1. Step 1: Identify the components. Radius (r) = 3.5 cm. Total height = 15.5 cm. Height of hemisphere = r = 3.5 cm. Height of cone (h) = 15.5 - 3.5 = 12 cm.

  2. Step 2: Find the slant height (l) of the cone. l = r2+h2\sqrt{r² + h²} = 3.52+122\sqrt{3.5² + 12²} = 12.25+144\sqrt{12.25 + 144} = 156\sqrt{156}.25 = 12.5 cm

  3. Step 3: Calculate CSA of hemisphere = 2πr² = 2 × (22/7) × 3.5×3.53.5 \times 3.5 = 77 cm²

  4. Step 4: Calculate CSA of cone = πrl = (22/7) × 3.5×12.53.5 \times 12.5 = 137.5 cm²

  5. Step 5: Total surface area of the toy = CSA of hemisphere + CSA of cone = 77 + 137.5 = 214.5 cm²

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