Q2:

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $14$ cm and the total height of the vessel is $13$ cm. Find the inner surface area of the vessel.

Given:

1. The vessel consists of a hollow hemisphere surmounted by a hollow cylinder.

2. The diameter of the hemisphere ($d$) = $14$ cm.

3. The total height of the vessel ($H$) = $13$ cm.

To Find:

The inner surface area of the vessel.

Step 1: Determine the dimensions of the hemisphere and cylinder.

Since the hemisphere is surmounted by the cylinder, the radius of the cylinder is equal to the radius of the hemisphere.

Radius ($r$) = $\frac{\text{Diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm}$.

The height of the hemisphere is equal to its radius, $r = 7$ cm.

Let the height of the cylindrical part be $h$.

Total height ($H$) = Height of cylinder ($h$) + Height of hemisphere ($r$).

$13 \text{ cm} = h + 7 \text{ cm}$.

$h = 13 \text{ cm} - 7 \text{ cm} = 6 \text{ cm}$.

Step 2: Identify the formula for the inner surface area.

The inner surface area of the vessel is the sum of the Curved Surface Area (CSA) of the cylinder and the Curved Surface Area (CSA) of the hemisphere.

Formula for CSA of cylinder = $2\pi rh$.

Formula for CSA of hemisphere = $2\pi r^2$.

Total Inner Surface Area = $2\pi rh + 2\pi r^2 = 2\pi r(h + r)$.

Step 3: Calculate the surface area.

Using $\pi = \frac{22}{7}$:

Total Inner Surface Area = $2 \times \frac{22}{7} \times 7 \times (6 + 7)$

[Canceling the 7 in the numerator and denominator]

Total Inner Surface Area = $2 \times 22 \times (13)$

Total Inner Surface Area = $44 \times 13$

Calculation: $44 \times 10 = 440$; $44 \times 3 = 132$; $440 + 132 = 572$.

Final Answer: The inner surface area of the vessel is 572 cm².