Question

The length of x of a rectangle is decreasing at the rate of 5 cm/min and width y is increasing at the rate of 4cm/min.When x = 8cm and y = 6cm find the rate of the perimeter and area of rectangle

Answer 1

Step 1:
Rate of decrease in length is dxdt=−5cm/min.
Negative sign shows it is decreasing.
Rate of increase in width is dydt=4cm/min.
Perimeter of the rectangle is P=2x+2y
Differentiating w.r.t t on both sides we get,
dpdt=2.dxdt+2.dydt
Step 2:
Now substituting the values for dxdt and dydt we get,
dpdt=2(−5)+2(4)
=−2cm/min.
Hence the perimeter decreases at the rate of −2cm/min.
Step 3:
Area of the rectangle is xy
A=xy
Differentiate on both sides w.r.t t
ddt(uv)=vdudt+u.dvdt
dAdt=x.dydt+y.dxdt
Substituting the values for x,y,dydt and dxdt is
dAdt=8×(4)+4×(−5)
=32−30
=2cm2/min
Hence the area of the rectangle increases at the rate of 2cm2/min

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