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A sheet of area 40m^2 is used to make an...

A sheet of area `40m^2` is used to make an open tank with square base. Find the dimensions of the base such that the volume of this tank is maximum.

Text Solution

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Let the length of base be x meters and height be y meters.

volume V=`x^(2)y`
Again x and y are related to the surface area of this tank which is equal to `40 m^(2)` Thus
`x^(2)+4xy+=40`
or `y=(40-x^(2))/(4x),x in 0,sqrt(400)`
`therefore V(x)=x^(2)(40-x^(2))/(4x)=(40x-x^(3))/(4)`
maximizing volume,
Let `V(x)=(40-3x^(2))/(4)=0 "or" x=sqrt(40)/(3)m`
and `V(x)=-(3x)/(2) "or" V sqrt(40)/(3)lt0`
Therefore volume is maximum at `x=sqrt(40)/(3)m`
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