Question

From a solid cylinder whose height is 2.8 cm and radius 2.1 cm, a conical cavity of the same height and same radius is hollowed out. Find the volume and the total surface area of the remaining solid.

Hint:

Be careful when calculating the surface area of a hollowed-out shape: hollowing out a cavity increases the total surface area because the inner conical wall is now exposed!

Answer 1

Step 1: Understanding the Question:
We are given a solid cylinder. A cone of the same dimensions is hollowed out of it. We need to calculate the remaining volume and the total surface area of the resulting solid shape.

Step 2: Key Formula or Approach:
- Cylinder and Cone dimensions: Radius \(R = 2.1 \text{ cm}\), Height \(H = 2.8 \text{ cm}\).
- Volume of remaining solid:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = \frac{2}{3} \pi R^2 H \]
- Total Surface Area (TSA) of remaining solid:
\[ \text{TSA} = \text{CSA of Cylinder} + \text{Area of base} + \text{CSA of Cone} \]
\[ \text{TSA} = 2\pi RH + \pi R^2 + \pi Rl \]
where \(l = \sqrt{R^2 + H^2}\) is the slant height of the cone.

Step 3: Detailed Explanation:

• Calculate Remaining Volume:
\[ V = \frac{2}{3} \times \frac{22}{7} \times (2.1)^2 \times 2.8 \]
\[ V = \frac{2}{3} \times \frac{22}{7} \times 4.41 \times 2.8 \]
\[ V = 2 \times 22 \times 0.21 \times 2.8 = 25.872 \text{ cm}^3 \]

• Calculate Slant Height (\(l\)) of the Cone:
\[ l = \sqrt{(2.1)^2 + (2.8)^2} = \sqrt{4.41 + 7.84} = \sqrt{12.25} = 3.5 \text{ cm} \]

• Calculate Total Surface Area (TSA):
- Curved Surface Area of Cylinder:
\[ \text{CSA}_{\text{cyl}} = 2\pi RH = 2 \times \frac{22}{7} \times 2.1 \times 2.8 = 36.96 \text{ cm}^2 \]
- Base Area of Cylinder:
\[ \text{Area}_{\text{base}} = \pi R^2 = \frac{22}{7} \times 2.1^2 = 13.86 \text{ cm}^2 \]
- Curved Surface Area of Cone:
\[ \text{CSA}_{\text{cone}} = \pi Rl = \frac{22}{7} \times 2.1 \times 3.5 = 23.1 \text{ cm}^2 \]
- Total Surface Area of remaining solid:
\[ \text{TSA} = 36.96 + 13.86 + 23.1 = 73.92 \text{ cm}^2 \]


Step 4: Final Answer:
The volume of the remaining solid is 25.872 \(\text{cm}^3\), and its total surface area is 73.92 \(\text{cm}^2\).