Multiple choice

The length, breadth and height of a rectangular solid are in the ratio $5: 4: 2$. If the total surface area is $1216\ cm^2,$ find the length, the breadth and the height of the solid.

  1. $20cm, 16cm$ and $ 8 cm$  
  2. $10cm, 16cm$ and $ 8 cm$  
  3. $10cm, 8cm$ and $ 2 cm$  
  4. $15cm, 12cm$ and $ 6cm$  
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A Correct answer
Explanation

Let dimensions be 5x, 4x, 2x. Surface area = 2(5x*4x + 4x*2x + 5x*2x) = 2(20x^2 + 8x^2 + 10x^2) = 2(38x^2) = 76x^2. 76x^2 = 1216, so x^2 = 16, x = 4. Dimensions are 20, 16, 8.

AI explanation

Let the length, breadth, and height be 5x, 4x, and 2x respectively. The total surface area formula for a rectangular solid is 2*(lb + bh + hl), which gives 2*((5x*4x) + (4x*2x) + (2x*5x)) = 2*(20x^2 + 8x^2 + 10x^2) = 76x^2. Setting this equal to the given area, 76x^2 = 1216, which simplifies to x^2 = 16, so x is 4. Multiplying the ratios by 4 gives the dimensions as 20 cm, 16 cm, and 8 cm. The result is 20cm, 16cm and 8cm.

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