. . Related rates A related rates problem is a problem which involves at least two changing quantities and asks you to figure out the rate at which one is changing given sufficient information on all of the others. For instance, when two vehicles drive in different directions we should be able to deduce the speed at which they are separating if we know their individual speeds and directions.
Example . If we blow air into a balloon of spherical shape at a rate of cm per second, at what rate the radius of the baloon changes when the radius is 7cm? Also compute the rate at which the surface area changes. The volume of the baloon of radius r is V = We are given dV dt = and we need to find dr dt when r = .
Now, dV dt = × × π r dr dt . Substituting r = and dV dt = , we get × × dr dt . Hence, dr dt = × × π . The surface area S of the baloon is S = .
Therefore, dS dt dr dt × × π Substituting dr dt = π and r = , we get dS dt = × × Therefore, the rate of change of radius is p cm/sec and the rate of change of surface area is cm / sec. Fig. . - - Applications of Differential Calculus Example .
The price of a product is related to the number of units available (supply) by the equation Px P , where P is the price of the product per unit in Rupees( ` ) and x is the number of units. Find the rate at which the price is changing with respect to time when units are available and the supply is increasing at a rate of units/week. We have, P = Therefore,