Chapter 1.9, Problem 90E

a.

To determine

Find the inverse function.

Answer to Problem 90E

The variable $x$ in the above function represents the exhaust temperature in degrees Fahrenheit and $y$ represents the percent load for a diesel engine.

Explanation of Solution

Calculation:

We are given exhaust temperature y in degrees Fahrenheit in terms of percent load for a diesel engine $x$ as

  $\begin{array}{l}y=0.03x^2+245.50,0<x<\mathrm{100.....}(1)\\ let\\ f\left(x\right)=y\\ f\left(x\right)=0.03x^2+245.50\\ replace,f\left(x\right)with,y\\ y=0.03x^2+245.50\\ x=0.03y^2+245.50\end{array}$

Finally solve for $y$ and replace $y$ by $f^{-1}\left(x\right)$

  $\begin{array}{l}0.03y^2=x-245.50y^2=\frac{x-245.50}{0.03}\\ y=\sqrt{\frac{x-245.50}{0.03}}\\ f^{-1}\left(x\right)=\sqrt{\frac{x-245.50}{0.03}}\\ =\sqrt{33.33x-8183.33}\end{array}$

Hence, the variable $x$ in the above function represents the exhaust temperature in degrees Fahrenheit and $y$ represents the percent load for a diesel engine.

b.

To determine

Graph the inverse function.

Answer to Problem 90E

  

Explanation of Solution

Calculation:

We are given exhaust temperature y in degrees Fahrenheit in terms of percent load for a diesel engine $x$ as

  $\begin{array}{l}y=0.03x^2+245.50,0<x<\mathrm{100.....}(1)\\ let,f\left(x\right)=y\end{array}$

The inverse function $y=\sqrt{33.33x-8183.33}$ is plotted below

  

c.

To determine

Find the percent load interval, temperature of the engine not exceed $500$ degrees.

Answer to Problem 90E

  $0<x<92.1$

Explanation of Solution

Calculation:

We are given exhaust temperature y in degrees Fahrenheit in terms of percent load for a diesel engine $x$ as

  $\begin{array}{l}y=0.03x^2+245.50,0<x<\mathrm{100.....}(1)\\ let,f\left(x\right)=y\end{array}$

From equation (1)

  $y<500$ or we can write as below

  $\begin{array}{l}0.03x^2+245.50\le 500,0<x<100\\ 0.03x^2\le 500-245.50,0<x<100\\ 0.03x^2\le 254.50,0<x<100\\ x^2\le \frac{254.50}{0.03},0<x<100\\ x\le \pm 92.105,0<x<100\\ x\le 92.105,0<x<100\end{array}$

Hence, the percent load interval is, $0<x<92.1$