Question

The dimensions of a rectangular box are in the ratio of $2:3:4$ and the difference between the cost of covering it with sheet of paper at the rates of $Rs.8$ and $Rs.9.50$ per $m^2$ is $Rs.1248.$ Find the dimensions of the box.

Answer 1

Ratio in the dimensions of a cuboidal box$=2:3:4$

Let length $\left(l\right)=4x$

Breadth $\left(b\right)=3x$

and height $\left(h\right)=2x$

$\therefore$ Total surface area$=2[lb+bh+hl]$

$=2[4x\times 3x+3x\times 2x+2x\times 4x]c{m}^2$

$=2[12x^2+6x^2+8x^2]=2\times 26x^2$

$=52x^2$

Now, rate of covering it with paper$=Rs8per{m}^2$

Therefore, cost of covering it with paper $=8\times 52x^2$

Also, rate of covering it with paper$=Rs9.50per{m}^2$

Therefore, cost of covering it with paper $=9.50\times 52x^2$

$\therefore$ Difference in the cost,

$=Rs(52x^2\times 9.50-52x^2\times 8)$

$=Rs(494x^2-416x^2)$

$=78x^2$

Now, according to the question,

$\Rightarrow 78x^2=1248$

$\Rightarrow x^2=\frac{1248}{78}=16$

$\Rightarrow x=\sqrt{16}=4$

$\therefore$ Dimensions are,

$h=2\times 4=8m$

$b=3\times 4=12m$

$l=4\times 4=16m$