The sum of length, breadth and depth of a cuboid is 19 cm and its principal diagonal is $5\sqrt{5}$ cm, then its surface area is :

Question

Accepted Answer

To determine the surface area of the cuboid, we need to use the given information: the sum of the length, breadth, and depth is 19 cm, and the principal diagonal is $5\sqrt{5}$ cm.

Let's denote the length, breadth, and depth of the cuboid as $l$, $b$, and $h$ respectively. According to the problem:

1. $l + b + h = 19$ cm
2. The principal diagonal of the cuboid is given by the formula $\sqrt{l^2 + b^2 + h^2} = 5\sqrt{5}$ cm.

From the second equation, we have:
$$
l^2 + b^2 + h^2 = (5\sqrt{5})^2 = 125
$$

We have two equations:
1. $l + b + h = 19$
2. $l^2 + b^2 + h^2 = 125$

To find the surface area, we use the formula for the surface area of a cuboid:
$$
\text{Surface Area} = 2(lb + bh + hl)
$$

To find $lb + bh + hl$, we use the identity:
$$
(l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl)
$$

Substituting the known values:
$$
19^2 = 125 + 2(lb + bh + hl)
$$
$$
361 = 125 + 2(lb + bh + hl)
$$
$$
236 = 2(lb + bh + hl)
$$
$$
lb + bh + hl = 118
$$

Thus, the surface area is:
$$
2(lb + bh + hl) = 2 \times 118 = 236 \, \text{cm}^2
$$

Therefore, the correct surface area of the cuboid is 236 cm$^2$, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1 (361 cm$^2$)**: This value is incorrect because it represents the square of the sum of the dimensions, not the surface area.
- **Option 2 (286 cm$^2$)**: This value does not satisfy the derived equation for the surface area based on the given conditions.
- **Option 4 (340 cm$^2$)**: This value is also incorrect as it does not match the calculated surface area using the provided dimensions and diagonal.

Hence, Option 3 is the correct answer, as it accurately reflects the calculated surface area of the cuboid based on the given conditions.