The sum of the length, breadth and height of a cuboid is p cm and its diagonal is q cm. Find its total surface

Question

The sum of the length, breadth and height of a cuboid is p cm and its diagonal is q cm. Find its total surface area (in cm 2 ).

Examoogle · Accepted Answer

The correct answer is Option B: $p^2 - q^2$. Recognize that the total surface area (TSA) is exactly the term $2(lb+bh+hl)$ in the expansion of $(l+b+h)^2$. Since $l^2+b^2+h^2 = q^2$, the result is simply $p^2 - q^2$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Recognize that the total surface area (TSA) is exactly the term $2(lb+bh+hl)$ in the expansion of $(l+b+h)^2$. Since $l^2+b^2+h^2 = q^2$, the result is simply $p^2 - q^2$.

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

Given: Length, breadth, and height sum (l+b+h) = p cm, and diagonal of the cuboid = q cm.

Steps: Define given variables: Let the length, breadth, and height be $l$, $b$, and $h$. The sum is given as $l+b+h = p$, and the diagonal formula is $q = \sqrt{l^2+b^2+h^2}$.. Apply algebraic identity: Square the sum of the dimensions to introduce the total surface area term $2(lb+bh+hl)$.. Substitute and solve: Substitute $p$ for the sum and $q^2$ for the sum of squares. The total surface area is $2(lb+bh+hl)$.

Answer: B is correct because the identity $(l+b+h)^2 = l^2+b^2+h^2+2(lb+bh+hl)$ simplifies directly to $p^2 = q^2 + TSA$, giving $TSA = p^2 - q^2$.

Exam Tip: This identity is a fundamental tool in coordinate geometry and vector algebra, where the squared magnitude of a sum of vectors depends on their dot products.

Examoogle · Answer

Option A: $(p - q)^2$

Examoogle · Answer

Option C: $\frac{p^2 - q^2}{2}$

Examoogle · Answer

Option D: $p^2 + q^2$

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Discussion (0)

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)