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MHT CET202310 May 2023Morning ShiftMathematicsApplication of DerivativesActual

An open metallic tank is to be constructed, with a square base and vertical sides, having volume 500 cubic meter. Then the dimensions of the tank, for minimum area of the sheet metal used in its construction, are

Options

  1. A5 ~m , 5 ~m , 10 ~m
  2. B10 ~m , 10 ~m , 5 ~m
  3. C2 ~m , 2 ~m , 8 ~m
  4. D15 ~m , 15 ~m , 5 ~m

Correct answer

B. 10 ~m , 10 ~m , 5 ~m

Step-by-step solution

Let the length, breadth and depth of open tank be x, x and y respectively. Volume ( V )=x^2 y 500=x^2 y ... (i) Total surface area of open tank is given by S =x^2+4 x y ... (ii) From (i), y= 500 x^2 From (ii), S =x^2+4 x 500 x^2 =x^2+ 2000 x Differentiating w.r.t. x , we get dS d x =2 x- 2000 x^2 For minimum area, dS d x =0 aligned & 2 x- 2000 x^2 =0 & 2000=2 x^3 & x^3=1000 aligned aligned & x=10 ~m & d ^2 ~S ~d x^2 =2+ 4000 x^3 & ( d ^2 ~S ~d x^2 )_ x=10 >0 aligned S is minimum when x=10 ~m and y=5 ~m ...[From (i)

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