Question

Find the total surface area of the given figure where $AB = 3.5 \text{ cm}$ and height of the toy $CD = 5 \text{ cm.}$ 

Hint:

For solids made up of two or more shapes, the total surface area is obtained by adding their curved surface areas only (if the base is common).

Answer 1

Step 1: Understand the figure.
The given figure is a toy made up of a hemisphere surmounted on a cone. - Radius of both = $r = AB = 3.5 \, \text{cm}$ - Height of the cone = $h = CD = 5 \, \text{cm}$ Step 2: Find the slant height of the cone.
By Pythagoras theorem, \[ l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + 5^2} = \sqrt{12.25 + 25} = \sqrt{37.25} = 6.1 \, \text{cm (approx.)} \] Step 3: Formula for total surface area (TSA).
\[ \text{TSA} = \text{Curved Surface Area of Cone} + \text{Curved Surface Area of Hemisphere} \] \[ \text{TSA} = \pi r l + 2\pi r^2 \]
Step 4: Substitute the values.
\[ \text{TSA} = \pi (3.5)(6.1) + 2\pi (3.5)^2 \] \[ = \pi (21.35 + 24.5) = \pi (45.85) \] \[ \text{TSA} = 45.85 \times 3.14 = 143.99 \, \text{cm}^2 \]
Step 5: Conclusion.
\[ \boxed{\text{Total Surface Area} = 144 \, \text{cm}^2 \, (\text{approx.})} \]