The sum of length, breadth and depth of a cuboid is 19 cm and the length of its diagonal is 11 cm. The surface area of the cuboid 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 length of the diagonal is 11 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 diagonal of the cuboid is given by the formula: $ \sqrt{l^2 + b^2 + h^2} = 11 $ cm

From the second equation, we have:
$$ l^2 + b^2 + h^2 = 11^2 = 121 $$

We need to find the surface area of the cuboid, which is given by:
$$ \text{Surface Area} = 2(lb + bh + hl) $$

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

Substituting the known values:
$$ 19^2 = 121 + 2(lb + bh + hl) $$
$$ 361 = 121 + 2(lb + bh + hl) $$
$$ 361 - 121 = 2(lb + bh + hl) $$
$$ 240 = 2(lb + bh + hl) $$

Thus, the surface area of the cuboid is:
$$ \text{Surface Area} = 240 \, \text{cm}^2 $$

Therefore, the correct answer is Option 2: $ 240 \, \text{cm}^2 $.

**Explanation for Incorrect Options:**

1. **Option 1: $120~cm^{2}$** - This option is incorrect because the calculations show that the surface area is 240 cm², not 120 cm². The value 120 cm² would imply a miscalculation in the identity or the arithmetic operations.

2. **Option 3: $360~cm^{2}$** - This option is incorrect because it overestimates the surface area. The correct calculation using the given conditions results in 240 cm², not 360 cm².

3. **Option 4: $140~cm^{2}$** - This option is incorrect because it underestimates the surface area. The correct calculation shows that the surface area is 240 cm², not 140 cm².

In conclusion, the calculations and logical reasoning confirm that the surface area of the cuboid is indeed 240 cm², making Option 2 the correct choice.