The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When 8 cm and y = 6 cm, find the rate of change of (a) the perimeter and (b) the area of the rectangle

Solution:

In maths, derivatives have wide usage.

They are used in many situations like finding maxima or minima of a function, finding the slope of the curve, and even inflection points.

It is given that dx/dt

= - 5 cm/minute,

dx/dt = 4 cm/minute, x = 8 cm and y = 6 cm

(a) The perimeter of a rectangle is given by

P = 2 (x + y)

Therefore,

dP/dt = 2 (dx/dt + dy/dt)

= 2 (- 5 + 4)

= - 2 cm/minute

(b) The area of a rectangle is given by

A = xy

Therefore,

dA/dt = dx/dt y + x dy/dt

= - 5 y + 4x

When, x = 8 cm and y = 6 cm

Then,

dA/dt = (- 5 × 6 + 4 x 8) cm²/minute

= 2 cm²/minute

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.1 Question 7

The length x of a rectangle is decreasing at the rate of 5cm / minute and the width y is increasing at the rate of 4 cm/minute. When 8 cm and y = 6 cm, find the rate of change of (a) the perimeter and (b) the area of the rectangle

Summary:

Given that the length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute.the rate of change of perimeter is -2 cm/ minute and the rate of change of area of the rectangle is 2 cm²/minute