Q9:

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 $10$ cm, and its base is of radius $3.5$ cm, find the total surface area of the article.

Given:

• Height of the solid cylinder ($h$) = $10$ cm
• Radius of the base of the cylinder ($r$) = $3.5$ cm
• The article is formed by scooping out a hemisphere from each end of the cylinder.

To Find:

The total surface area of the resulting wooden article.

Step 1: Understanding the Surface Area Components

The total surface area of the article consists of three parts:

1. The Curved Surface Area (CSA) of the cylinder.
2. The Curved Surface Area (CSA) of the top hemisphere.
3. The Curved Surface Area (CSA) of the bottom hemisphere.

Formulae to be used:

• CSA of cylinder = $2\pi rh$
• CSA of a hemisphere = $2\pi r^2$

Step 2: Setting up the Equation

Total Surface Area (TSA) = (CSA of cylinder) + (CSA of top hemisphere) + (CSA of bottom hemisphere)

$TSA = 2\pi rh + 2\pi r^2 + 2\pi r^2$

$TSA = 2\pi rh + 4\pi r^2$

$TSA = 2\pi r(h + 2r)$

Step 3: Substituting the Values

Given $r = 3.5$ cm and $h = 10$ cm.

$TSA = 2 \times \frac{22}{7} \times 3.5 \times (10 + 2 \times 3.5)$

[Since $\pi \approx \frac{22}{7}$]

Step 4: Performing the Calculation

First, simplify the radius term: $3.5 = \frac{7}{2}$

$TSA = 2 \times \frac{22}{7} \times \frac{7}{2} \times (10 + 7)$

$TSA = 2 \times \frac{22}{7} \times \frac{7}{2} \times (17)$

Cancel the common factors ($2$ and $7$):

$TSA = 22 \times 17$

$TSA = 374$

Step 5: Final Conclusion

The total surface area is calculated in square centimeters ($cm^2$).

Final Answer: 374 cm²