A cuboid has a total surface area (TSA) of 94 sq. cm and a volume of 60 cubic cm. Its length is $l$ cm, breadth is $b$ cm, and height is $h$ cm, where $l$, $b$, and $h$ are positive consecutive integers such that $l

Question

A cuboid has a total surface area (TSA) of 94 sq. cm and a volume of 60 cubic cm. Its length is $l$ cm, breadth is $b$ cm, and height is $h$ cm, where $l$, $b$, and $h$ are positive consecutive integers such that $l < b < h$.

Compare the following:

**Quantity I:** Lateral Surface Area (LSA) of the cuboid.

**Quantity II:** Total Surface Area of a cube with side length $b$.

Which of the following is true?

Accepted Answer

**Solution:**

1. Given the cuboid has TSA = 94 sq. cm and volume = 60 cubic cm.

2. The dimensions $l$, $b$, and $h$ are positive consecutive integers with $l < b < h$.

3. We need to find integers $l$, $b$, and $h$ such that:
   - $l 	imes b 	imes h = 60$
   - $2(lb + bh + hl) = 94$

4. Try $l = 3$, $b = 4$, $h = 5$:
   - Volume = $3 	imes 4 	imes 5 = 60$ (matches)
   - TSA = $2(3 	imes 4 + 4 	imes 5 + 5 	imes 3) = 2(12 + 20 + 15) = 2 	imes 47 = 94$ sq. cm (matches)

5. Calculate Quantity I (LSA of cuboid):
   - LSA = $2h(l + b) = 2 	imes 5 	imes (3 + 4) = 2 	imes 5 	imes 7 = 70$ sq. cm

6. Calculate Quantity II (TSA of cube with side $b=4$):
   - TSA = $6 	imes b^2 = 6 	imes 4^2 = 6 	imes 16 = 96$ sq. cm

7. Compare:
   - Quantity I = 70
   - Quantity II = 96

Since $70 < 96$, Quantity I < Quantity II.

**Answer:** Quantity I < Quantity II

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