Question

A cylinder of radius \(r\) is surmounted on a hemisphere of same radius. If total height of the object is 13 cm, then its inner surface area is

• A. \(2\pi r(r + 13)\)
• B. \(13\pi r\)
• C. \(2\pi(13 + r)^2\)
• D. \(26\pi r\)

Hint:

Notice how the terms involving \(r^2\) cancel out beautifully in the final step.
This reveals that the surface area of such a combined shape depends only on the radius and the total height \(H\), simplifying to \(2\pi r H\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem describes a composite solid consisting of a cylinder of radius \(r\) mounted on top of a hemisphere of the same radius \(r\).
The total height of this combined object is given as \(13\text{ cm}\). We need to determine the total inner surface area of this solid.

Step 2: Key Formula or Approach:
1. Let \(r\) be the radius of both the cylinder and the hemisphere.
2. Let \(H\) be the total height of the object, where \(H = 13\text{ cm}\).
3. The height of the cylindrical portion, \(h\), is obtained by subtracting the radius of the hemisphere from the total height: \[ h = H - r = 13 - r \] 4. The inner surface area of the solid is the sum of the curved surface area of the cylinder and the curved surface area of the hemisphere: \[ \text{Total Inner Surface Area (S)} = \text{Curved Surface Area of Cylinder} + \text{Curved Surface Area of Hemisphere} \] \[ S = 2\pi rh + 2\pi r^2 \]

Step 3: Detailed Explanation:
1. Write down the formulas for the individual surface areas: - Curved Surface Area of the Cylinder: \[ \text{CSA}_{\text{cylinder}} = 2\pi r h \] - Curved Surface Area of the Hemisphere: \[ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 \] 2. Express the height of the cylinder \(h\) in terms of total height \(H = 13\text{ cm}\) and radius \(r\): \[ h = 13 - r \] 3. Substitute \(h = 13 - r\) into the total surface area formula: \[ S = 2\pi r(13 - r) + 2\pi r^2 \] 4. Expand the expression: \[ S = 26\pi r - 2\pi r^2 + 2\pi r^2 \] 5. Simplify by cancelling the equal and opposite terms: \[ S = 26\pi r \]

Step 4: Final Answer:
The inner surface area of the combined solid is \(26\pi r\text{ cm}^2\).
Thus, the correct option is (D).