Chapter 7.4, Problem 82STP

To determine

Find the range of speeds that would give a gas mileage of atleast 25 miles per gallon.

Answer to Problem 82STP

The speed of the vehicle should be = $\left\{x\right|22.087\le x\le 67.913\}$ miles per hour

Explanation of Solution

Given:

The gas mileage y in miles per gallon for a certain vehicle is given by the equation $y=10+0.9x-0.01x^2$ ,where x is the speed of the vehicle between 10 and 75 miles per hour.

Calculation:

Given:

  $y=10+0.9x-0.01x^2$

y is atleast 25 miles per gallon.

So,

  $\begin{array}{l}y\ge 25\\ \Rightarrow y=10+0.9x-0.01x^2\ge 25\\ \Rightarrow -0.01x^2+0.9x-15\ge \mathrm{0.............}[\text{subtract 25 from each side]}\end{array}$

To solve the inequality , graph the related function $-0.01x^2+0.9x-15=0$

  

Since the graph opens downwards and intersects x-axis at x = 22.087 and x = 67.913

Also, we need $y\ge 0$.

The graph lies above the x- axis for the values of x : $\left\{x\right|22.087\le x\le 67.913\}$

So, the speed of the vehicle should be = $\left\{x\right|22.087\le x\le 67.913\}$ miles per hour