Surface Areas and Volumes | Exercise 12.1

Question 9

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is $10\text{ cm}$ , and its base is of radius $3.5\text{ cm}$ , find the total surface area of the article.

Solution

Understand the Question

When a hemisphere is scooped out from each end of a solid cylinder, the flat circular ends are replaced by curved hemispherical cavities.
Therefore, the total surface area of the resulting article is the sum of the curved surface area (CSA) of the cylinder and the curved surface area of the two hemispheres: $\text{Total Surface Area} = \text{CSA of cylinder} + 2 \times (\text{CSA of hemisphere}) = 2\pi rh + 4\pi r^2$
Given values:
- Radius, $r = 3.5\text{ cm}$
- Height of cylinder, $h = 10\text{ cm}$

Step 1 · Calculate Cylinder's Curved Surface Area

Given: Height of cylinder, $h = 10\text{ cm}$ Radius of base, $r = 3.5\text{ cm}$ Diagram 1

\begin{aligned} \text{CSA of cylinder} &= 2\pi rh \\[0.6em] &= 2 \times \dfrac{22}{7} \times 3.5 \times 10 \\[0.6em] &= 2 \times 22 \times 0.5 \times 10 \\[0.6em] &= 220\text{ cm}^2 \end{aligned}

Step 2 · Calculate Hemispheres' Curved Surface Area

Radius of each hemisphere, $r = 3.5\text{ cm}$

\begin{aligned} \text{CSA of 2 hemispheres} &= 2 \times (2\pi r^2) \\[0.6em] &= 4\pi r^2 \\[0.6em] &= 4 \times \dfrac{22}{7} \times (3.5)^2 \\[0.6em] &= 4 \times \dfrac{22}{7} \times 3.5 \times 3.5 \\[0.6em] &= 4 \times 22 \times 0.5 \times 3.5 \\[0.6em] &= 154\text{ cm}^2 \end{aligned}

Step 3 · Calculate Total Surface Area

Sum the curved surface areas:

\begin{aligned} \text{Total Surface Area} &= \text{CSA of cylinder} + \text{CSA of 2 hemispheres} \\[0.6em] &= 220 + 154 \\[0.6em] &= 374\text{ cm}^2 \end{aligned}

Answer

$374\text{ cm}^2$

Common Mistakes

Subtracting Surface Area: Scooping out material decreases volume, but it increases surface area by exposing new inner curved surfaces. Do not subtract the hemisphere area.
Including Flat Base Areas: The flat circular ends are scooped away, so do not add $\pi r^2$ for the bases.

Question 9

More questions in Exercise 12.1