The ratio of the length, width and height of a closed cuboid is given as $6 : 3 : 2$. The total surface area of this cuboid is given as $1800 ext{ cm}^2$. Find the volume (in $ ext{cm}^3$) of this cuboid.

Question

The ratio of the length, width and height of a closed cuboid is given as $6 : 3 : 2$. The total surface area of this cuboid is given as $1800 	ext{ cm}^2$. Find the volume (in $	ext{cm}^3$) of this cuboid.

Accepted Answer

**Solution:**

1. Given the ratio of length, breadth, and height as $6 : 3 : 2$, let the dimensions be:
   - Length, $l = 6x$
   - Breadth, $b = 3x$
   - Height, $h = 2x$

2. The total surface area (TSA) of a cuboid is given by:
   $$	ext{TSA} = 2(lb + bh + hl)$$

3. Substitute the values:
   $$2[(6x)(3x) + (3x)(2x) + (2x)(6x)] = 1800$$
   $$2(18x^2 + 6x^2 + 12x^2) = 1800$$
   $$2 	imes 36x^2 = 1800$$
   $$72x^2 = 1800$$

4. Solve for $x^2$:
   $$x^2 = \frac{1800}{72} = 25$$
   $$x = 5$$

5. Calculate the actual dimensions:
   - Length, $l = 6 	imes 5 = 30 	ext{ cm}$
   - Breadth, $b = 3 	imes 5 = 15 	ext{ cm}$
   - Height, $h = 2 	imes 5 = 10 	ext{ cm}$

6. Calculate the volume:
   $$	ext{Volume} = l 	imes b 	imes h = 30 	imes 15 	imes 10 = 4500 	ext{ cm}^3$$

**Answer:** $4500$

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