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Question

If length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2 cm/minute, when x=10 cm and y=6cm, find the rates of change of (i) the perimeter, (ii) the area of the rectangle.

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Solution

Rate of decrease in length is dxdt=3cm/min
Negative sign shows it is decreasing.
Rate of increase in width is dydt=2cm/min
Perimeter of the rectangle is P=2x+2y
Differentiating w.r.t t on both sides we get,
dpdt=2dxdt+2dydt

Now substituting the values for dxdt and dydt we get,
dPdt=2(3)+2(2)
=2cm/min
Hence the perimeter decreases at the rate of 2cm/min.

Area of the rectangle is xy
A=xy
Differentiate on both sides w.r.t t, we get
ddt(uv)=vdudt+udvdt
dAdt=xdydt+ydxdt
Substituting the values for x,y,dydt and dxdt is
dAdt=10×(2)+6×(3)
=2018=2cm2/min
Hence the area of the rectangle increases at the rate of 2cm2/min

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