The sum of the length, the breadth and the height of a cuboid is $ 38 ext{ cm}$ and the length of its diagon

Question

The sum of the length, the breadth and the height of a cuboid is $ 38 	ext{ cm}$ and the length of its diagonal is $ 22 	ext{ cm}$. Find the total surface area of the cuboid.

Examoogle · Accepted Answer

The correct answer is Option C: $960 	ext{ cm}^2$. Use the algebraic identity $(l+b+h)^2 = (l^2+b^2+h^2) + TSA$. Square the sum of dimensions (38) and subtract the square of the diagonal (22).

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the algebraic identity $(l+b+h)^2 = (l^2+b^2+h^2) + TSA$. Square the sum of dimensions (38) and subtract the square of the diagonal (22).

Formula Used: $$(l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + lh)$$

Given: Sum of length, breadth, and height (l + b + h) = 38 cm; length of the diagonal (d) = 22 cm.

Steps: Define given parameters: Let the dimensions be $l$, $b$, and $h$. We are given $l + b + h = 38$ and the diagonal $d = \sqrt{l^2 + b^2 + h^2} = 22$.. Calculate the sum of squares: Square the diagonal equation to find the value of $l^2 + b^2 + h^2$.. Use algebraic identity for Total Surface Area (TSA): We know the identity $(l + b + h)^2 = (l^2 + b^2 + h^2) + 2(lb + bh + lh)$, where $2(lb + bh + lh)$ is the Total Surface Area (TSA).. Solve for TSA: Calculate $38^2 = 1444$ and subtract $484$ to isolate the TSA.

Answer: C is correct because the calculated Total Surface Area is 960 cm².

Exam Tip: This identity-based approach is frequently used in algebraic problems involving $(x+y+z)^2$ and in 3D geometry problems related to sphere or cuboid properties.

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