Question

The dimensions of a rectangular box are in the ratio $4:2:3$. The difference between cost of covering it with paper at Rs. $12$ per $m^2$ and with paper at the rate of $13.50$ per $m^2$ is Rs. $1,248$. Find the dimensions of the box.

• A. $6$ m, $3$ m and $12$ m
• B. $16$ m, $8$ m and $12$ m
• C. $11$ m, $7$ m and $12$ m
• D. $13$ m, $9$ m and $12$ m

Answer 1

The correct option is B $16$ m, $8$ m and $12$ m

Given, a rectangular box with dimension in ratio $4:2:3$.
Also, difference between cost of covering it with paper at rate Rs $12$ per sq m and at rate Rs $13.5$ per sq m is $1248$
Let the dimension of Cuboid be $4x,2x,3x$.
For covering it with paper we need to find area of paper $=$ Total surface area of cuboid
$=2(lb+bh+lb)=2(4x\times 2x+2x\times 3x+4x\times 3x)=2(8x^2+6x^2+12x^2)=2\left(26x^2\right)=52x^2$
Cost of covering it at rate Rs. $12$ per sq m$=52x^2\times 12=624x^2$
Cost of covering it at rate Rs. $13.5$ per sq m$=52x^2\times 13.5=704x^2$
Difference in the cost of two paper $=$ $1248$
$\Rightarrow 704x^2-624x^2=1248$
$\Rightarrow 78x^2=1248$
$\Rightarrow x^2=16$
$\Rightarrow x=4$
Dimensions of cuboid $=4x=4\times 4=16$ m
$=2x=2\times 4=8$ m
$=3x=3\times 4=12$ m