Question

The length of the diagonal of a cuboid is `13sqrt2` cm and its volume and total surface area is respectively 780 cm³ and 562 cm². Find the dimension of the cuboid.

Answer 1

Let the dimensions (length, breadth, and height) of the cuboid be denoted by l, b, and h respectively.

From the problem statement:

Length of the diagonal \[= \sqrt{l^2 + b^2 + h^2} = 13\sqrt{2}\text{ cm}\]

Volume = lbh = 780 cm³

Total Surface Area \[(\text{TSA}) = 2(lb + bh + hl) = 562\text{ cm}^2\]

Squaring the diagonal equation:

\[l^2 + b^2 + h^2 = (13\sqrt{2})^2 = 169 \times 2 = 338\]

We know the algebraic identity for the square of the sum of three variables:

\[(l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl)\]

Substitute the known values into the identity:

\[(l + b + h)^2 = 338 + 562\]

\[(l + b + h)^2 = 900\]

\[l + b + h = \sqrt{900} = 30\text{ cm}\]

Now, let's find the relations or values using the volume and surface area. We can test integer factors of the volume 780 to find the dimensions l, b, and h that satisfy l + b + h = 30 and \[l^2 + b^2 + h^2 = 338\].

Let's test factors of 780:

Consider dimensions 5, 12, and 13:

Sum: 5 + 12 + 13 = 30 (matches l + b + h = 30)

Sum of squares:\[5^2 + 12^2 + 13^2 = 25 + 144 + 169 = 338\] (matches \[l^2 + b^2 + h^2 = 338\])

Product (Volume): \[5 \times 12 \times 13 = 780\text{ cm}^3\] (matches the given volume)

Thus, the dimensions of the cuboid are 5 cm, 12 cm, and 13 cm.