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The length x of a rectangle is decreasing at therate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute.When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and(b) the area of the rectangle |
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Answer» Given that `(dx)/(dt) = -5 cm`/minute and `(dy)/(dt)` = 4 cm/minute. (i) Let P be the perimeter of the rectangle at any instant. Then, `P = 2(x +y) rArr (dP)/(dt) = 2 ((dx)/(dt) + (dy)/(dt)) = 2 (-5 + 4)` cm/minute Hence, the perimeter of the rectangle is decreasing at the rate of 2 cm/minute. (ii) Let A the area of the rectangle at any instant. Then, `A = x.y rArr (dA)/(dt) = (dx)/(dt) .y + x.(dy)/(dt)` `= [(-5) xx 6 + 8 xx4] cm^(2)`/minute `= 2 cm^(2)`/minute Hence the area of the rectangle is increasing at the rate of `2 cm^(2)`/minute. |
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