From a rectangular solid of metal 42 cm by 30 cm by 20 cm, a conical cavity of diameter 14 cm and depth 24 cm is drilled out. Find :
the surface area of remaining solid,

Answer

To find the surface area of the remaining solid after drilling a conical cavity from a rectangular solid, we can follow these steps:

### Step 1: Identify the dimensions of the rectangular solid.
- Length (l) = 42 cm
- Breadth (b) = 30 cm
- Height (h) = 20 cm

### Step 2: Calculate the surface area of the rectangular solid.
The surface area (SA) of a rectangular solid (cuboid) can be calculated using the formula:
\[ SA = 2(lb + bh + lh) \]
Substituting the values:
\[ SA = 2(42 \times 30 + 30 \times 20 + 20 \times 42) \]
Calculating each term:
- \(42 \times 30 = 1260\)
- \(30 \times 20 = 600\)
- \(20 \times 42 = 840\)

Now, substituting these values back into the formula:
\[ SA = 2(1260 + 600 + 840) = 2(2700) = 5400 \text{ cm}^2 \]

### Step 3: Calculate the area of the base of the conical cavity.
The diameter of the conical cavity is given as 14 cm, so the radius (r) is:
\[ r = \frac{14}{2} = 7 \text{ cm} \]
The area of the circular base of the cone (A_circle) is given by:
\[ A_{circle} = \pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \text{ cm}^2 \]

### Step 4: Calculate the curved surface area (CSA) of the conical cavity.
To find the CSA of the cone, we first need to calculate the slant height (l) using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ cm} \]
Now, the CSA of the cone is given by:
\[ CSA_{cone} = \pi r l = \frac{22}{7} \times 7 \times 25 = 22 \times 25 = 550 \text{ cm}^2 \]

### Step 5: Calculate the total surface area of the remaining solid.
The surface area of the remaining solid consists of:
1. The surface area of the rectangular solid (without the circular base of the cone).
2. The CSA of the cone.
3. The area of the circular base of the cone is subtracted from the total surface area of the rectangular solid.

Thus, the surface area of the remaining solid is:
\[ SA_{remaining} = SA_{cuboid} - A_{circle} + CSA_{cone} \]
Substituting the values:
\[ SA_{remaining} = 5400 - 154 + 550 \]
Calculating this:
\[ SA_{remaining} = 5400 - 154 + 550 = 5400 - 154 + 550 = 5796 \text{ cm}^2 \]

### Final Answer:
The surface area of the remaining solid is \(5796 \text{ cm}^2\).

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