Chapter 5.2, Problem 91AE

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using $y=f\left(x\right)$ to represent a function, an applied problem might use $C=C\left(q\right)$ to represent the cost $C$ of manufacturing q units of a good. Because of this, the inverse notation $f^{-1}$ used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as $C=C\left(q\right)$ will be $q=q\left(C\right)$. So $C=C\left(q\right)$ is a function that represents the cost $C$ as a function of the number $q$ of units manufactured, and $q=q\left(C\right)$ is a function that represents the number $q$ as a function of the cost $C$. Problems 91-94 illustrate this idea.

Vehicle Stopping Distance Taking into account reaction time, the distance $d$ (in feet) that a car requires to come to a complete stop while traveling $r$ miles per hour is given by the function

$d\left(r\right)=6.97r-90.39$

(a) Express the speed $r$ at which the car is traveling as a function of the distance $d$ required to come to a complete stop.

(b) Verify that $r=r\left(d\right)$ is the inverse of $d=d\left(r\right)$ by showing that $r\left(d\left(r\right)\right)=r$ and $d\left(r\left(d\right)\right)=d$.

(c) Predict the speed that a car was traveling if the distance required to stop was 300 feet.