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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 m2 is Rs. 1248. Find the dimensions of the box.
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Solution
\[\text{ Suppose that the dimensions be x multiple of each other } . \]
\[\text { The dimensions are in the ratio 2: 3: 4 . } \]
\[\text { Hence, length = 2x m }\]
\[\text { Breadth = 3x }m\]
\[\text { Height = 4x m }\]
\[\text { So, total surface area of the rectangular box }= 2 \times \text { (length }\times \text { breadth + breadth } \times \text { height + length }\times \text { height) }\]
\[ = 2 \times (2x \times 3x + 3x \times 4x + 2x \times 4x)\]
\[ = 2 \times (6 x^2 + 12 x^2 + 8 x^2 )\]
\[ = 2 \times (26 x^2 )\]
\[ = 52 x^2 m^2 \]
\[\text { Also, the cost of covering the box with paper at the rate Rs }8/ m^2 \text { and Rs } 9 . 50/ m^2 is Rs 1248 . \]
\[\text { Here, the total cost of covering the box at a rate of Rs } 8/ m^2 = 8 \times 52 x^2 = Rs 416 x^2 \]
\[\text { And the total cost of covering the box at a rate of Rs 9 } . 50/ m^2 = 9 . 50 \times 52 x^2 = Rs 494 x^2 \]
\[\text { Now, total cost of covering the box at the rate Rs 9 . } 50/ m^2 - \text { total cost of covering the box at the rate Rs } 8/ m^2 = 1248\]
\[ \Rightarrow 494 x^2 - 416 x^2 = 1248\]
\[ \Rightarrow 78 x^2 = 1248\]
\[ \Rightarrow x^2 =\frac{1248}{78} = 16\]
\[ \Rightarrow x = \sqrt{16} = 4\]
\[\text { Hence, length of the rectangular box }= 2 \times x = 2 \times 4 = 8 m \]
\[\text { Breadth = 3 } \times x = 3 \times 4 = 12 m \]
\[\text { Height = 4 }\times x = 4 \times 4 = 16 m\]
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