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

Understand the Question The vessel is a combination of two hollow shapes: a cylinder mounted on top of a hemisphere. Both shapes share the same radius: . The height of the hemisphere equals its radius (), so the height of the cylinder is . Since the vessel is hollow and open, the inner surface area equals the sum of the curved surface areas: Step 1 · Find the Radius and Height of the Cylinder Given diameter of the hemisphere . Radius of hemisphere and cylinder: Height of the hemisphere . Height of the cylinder (): Step 2 · Calculate the Inner Surface Area Inner surface area of the vessel: Taking common and substituting , , and : Answer > Common Mistakes Using Total Height as Cylinder Height: Using directly instead of subtracting the hemisphere's radius (). Adding Base Areas: Using total surface area (TSA) formulas instead of curved surface area (CSA) for hollow objects, which incorrectly adds circular bases that do not exist inside the open vessel.

Surface Areas and Volumes | Exercise 12.1

Question 2

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

Solution

Understand the Question

The vessel is a combination of two hollow shapes: a cylinder mounted on top of a hemisphere.
Both shapes share the same radius: $r = \dfrac{14}{2} = 7\text{ cm}$ .
The height of the hemisphere equals its radius ( $7\text{ cm}$ ), so the height of the cylinder is $h = 13 - 7 = 6\text{ cm}$ .
Since the vessel is hollow and open, the inner surface area equals the sum of the curved surface areas: $\text{Inner Surface Area} = \text{CSA of Cylinder} + \text{CSA of Hemisphere} = 2\pi rh + 2\pi r^2$

Step 1 · Find the Radius and Height of the Cylinder

Given diameter of the hemisphere $= 14\text{ cm}$ .

Radius of hemisphere and cylinder: $r = \dfrac{\text{diameter}}{2} = \dfrac{14}{2} = 7\text{ cm}$ Diagram 1

Height of the hemisphere $= r = 7\text{ cm}$ .

Height of the cylinder ( $h$ ):

\begin{aligned} h &= \text{Total height} - \text{Radius} \\[0.6em] &= 13 - 7 \\[0.6em] &= 6\text{ cm} \end{aligned}

Step 2 · Calculate the Inner Surface Area

Inner surface area of the vessel: $\text{Inner surface area} = \text{CSA of cylinder} + \text{CSA of hemisphere}$

$\text{Inner surface area} = 2\pi rh + 2\pi r^2$

Taking $2\pi r$ common and substituting $\pi = \dfrac{22}{7}$ , $r = 7\text{ cm}$ , and $h = 6\text{ cm}$ :

\begin{aligned} &= 2\pi r(h + r) \\[0.6em] &= 2 \times \dfrac{22}{7} \times 7 \times (6 + 7) \\[0.6em] &= 2 \times 22 \times 13 \\[0.6em] &= 44 \times 13 \\[0.6em] &= 572\text{ cm}^2 \end{aligned}

Answer

$572\text{ cm}^2$

Common Mistakes

Using Total Height as Cylinder Height: Using $h = 13\text{ cm}$ directly instead of subtracting the hemisphere's radius ( $h = 13 - 7 = 6\text{ cm}$ ).
Adding Base Areas: Using total surface area (TSA) formulas instead of curved surface area (CSA) for hollow objects, which incorrectly adds circular bases that do not exist inside the open vessel.

Question 2

More questions in Exercise 12.1