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. 1248. Find the dimensions of the box.

A. 12 m, 9 m and 13 m
B. 16 m, 8 m and 12 m
C. 11 m, 7 m and 11 m
D. 19 m, 2 m and 9 m

Answer 1

Hint: Let the length, breadth and height of the rectangular box be 4x, 2x and 3x respectively and use the formula of total surface area to calculate the area which is given as follows:
Total surface area = 2 (lb + bl + hl), where l, b and h are length, breadth and height respectively of rectangular box.

Complete step-by-step answer:
We have been given the ratio of dimensions of a rectangular box as 4 : 2 : 3.
So let us suppose the length, breadth and height of the rectangular box to be 4x, 2x and 3x respectively.
We know the formula for finding total surface area = 2 (lb + bh + hl)
So we have, l = 4x, b = 2x and h = 3x. Therefore we can substitute these in the formula and we will get as follows:
\[\therefore \]Total surface area
\[\begin{align}
  & =2\left( 4x\times 2x+2x\times 3x+4x\times 3x \right) \\
 & =2\left( 8{{x}^{2}}+6{{x}^{2}}+12{{x}^{2}} \right) \\
 & =2\left( 26{{x}^{2}} \right) \\
 & =52{{x}^{2}}{{m}^{2}} \\
\end{align}\]
Now it is given in the question that the cost of covering the box of area 1 \[{{m}^{2}}\] with a paper is Rs. 12.
Hence we get the cost of covering the cox of area \[52{{x}^{2}}{{m}^{2}}=12\times 52{{x}^{2}} = Rs. 624{{x}^{2}} \]
Again, it is given to us that the cost of covering the box of area 1 \[{{m}^{2}}\] with another paper is Rs. 13.50.
Hence we get the cost of covering the box of area \[52{{x}^{2}}{{m}^{2}}=13.50\times 52{{x}^{2}} = Rs. 702{{x}^{2}} \]
Also, we have been given the difference between the cost is equal to Rs. 1248.
So we can formulate an equation as below:
\[\begin{align}
  & 702{{x}^{2}}-624{{x}^{2}}=1248 \\
 & 78{{x}^{2}}=1248 \\
 & {{x}^{2}}=\dfrac{1248}{78} \\
 & {{x}^{2}}=16 \\
\end{align}\]
Hence, \[x=4\].
Hence the values of the length, breadth and height are \[4\times 4=16,2\times 4=8m\] and \[3\times 4=12m\] respectively. Therefore, the correct option of the above question is option B.

Note: Be careful while calculating the surface area and don’t miss the ‘s’ which is outside the bracket of the formula. To find the value of x, we have to formulate the correct equation. If by mistake we add the costs instead of subtracting, then we will get the wrong answer.