Answer
$50$ miles/hour
Work Step by Step
Let's note by $x$ her speed from Tortula to Cactus. She drove $\dfrac{250}{x}$ hours from Tortula to Cactus and $\dfrac{360}{x+10}$ from Cactus to Dry Junction. We write the equation for the total time and solve it for $x$:
$$\begin{align*}
\dfrac{250}{x}+\dfrac{360}{x+10}&=11\\
250(x+10)+360x&=11x(x+10)\\
250x+2500+360x&=11x^2+110x\\
610x+2500&=11x^2+110x\\
11x^2+110x-610x-2500&=0\\
11x^2-500x-2500&=0\\
x&=\dfrac{-(-500)\pm\sqrt{(-500)^2-4(11)(-2500)}}{2(11)}\\
&=\dfrac{500\pm 600}{22}\\
x_1&=\dfrac{500-600}{22}=-\dfrac{50}{11}\\
x_2&=\dfrac{500+600}{22}=50.
\end{align*}$$
Because $x>0$, the only solution that fits is $x=50$ miles/hour.
