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A farmer F(1) has a land in the shape of...

A farmer `F_(1)` has a land in the shape of a triangle with vertices at `P(0,0),Q(1,1) and R(2,0).` From this land, a neighboring farmer `F_(2)` takes away the region which lies between the side PQ and curve of the from `y=x^(n)(ngt1).` If the area of the region taken away by the farmer `F_(2)` is exactly `30%` of the area of `DeltaPQR`, then the value of n is ___.

Text Solution

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The correct Answer is:
4
We have, ` y= x^(n), n gt 1 `
`because P(0,0)Q(1,1) and R(2,0) " are vertices of " triangle PQR.`

` therefore " Area of shaded region "=30% " of area of " triangle PQR `
`rArr int_(0)^(1)(x-x^(n))dx=(30)/(100)xx(1)/(2)xx 2 xx 1 `
` rArr [(x^(2))/(2)-(x^(n+1))/(n+1)]_(0)^(1)=(3)/(10) rArr ((1)/(2)-(1)/(n+1))=(3)/(10)`
` rArr (1)/(n+1)=(1)/(2)-(3)/(10)=(2)/(10)=(1)/(5) rArr n+1=5 rArr n=4 `
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