College Algebra, Chapter 1, 1.6, Section 1.6, Problem 92

The equation $g = 10 + 0.9 \nu - 0.01 \nu^2$ represents the gas mileage measured in $\displaystyle \frac{\text{mi}}{\text{gal}}$ for a particular vehicle, driven at $\displaystyle \nu \frac{\text{mi}}{\text{h}}$, as long as $\nu$ is between $\displaystyle 10 \frac{\text{mi}}{\text{h}}$ and $\displaystyle 75 \frac{\text{mi}}{\text{h}}$. For what range of speeds is the vehicle's mileage $\displaystyle 30 \frac{\text{mi}}{\text{gal}}$ or better?

$
\begin{equation}
\begin{aligned}
10 + 0.9 \nu - 0.01 \nu^2 & \geq 30 && \text{Model}\\
\\
-20 + 0.9 \nu - 0.01 \nu^2 & \geq 0 && \text{Subtract }30\\
\\
2000 - 90 \nu + \nu^2 & \leq 0 && \text{Divide both sides by } -0.01\\
\\
(\nu - 50 ) ( \nu - 40) & \leq 0 && \text{Factor out}
\end{aligned}
\end{equation}
$

We have, $\nu \leq 50$ and $\nu \leq 40$
It shows that if the range of speeds (the intersection of these inequalities) is $\nu \leq 40$, then the vehicles mileage is $\displaystyle 30 \frac{\text{mi}}{\text{gal}}$ or better.

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