Answer
She runs at 3.78 mi/h
Work Step by Step
$$v_{run} = v$$ $$v_{cycle} = v + 8$$
- Dividing the distance by the speed of each type of exercise, we will get the time (in hours). We know that the sum of both times is equal to 1 hour.
$$\frac 4{v_{cycle}} + \frac{2.5}{v_{run}} = 1$$ $$\frac 4{v + 8} + \frac{2.5}{v} = 1$$ $$\frac 4{v + 8} = 1 - \frac{2.5}{v}$$ $$ 4 = (v+8)(1 - \frac{2.5}{v}) = v + 8 - 2.5 - \frac{8 \times 2.5}{v}$$ $$4 = v + 5.5 - \frac{20}{v}$$ $$\frac{20} v = v + 5.5 - 4 = v + 1.5$$ $$20 = v^2 + 1.5v $$ $$0 = v^2 + 1.5v - 20$$ $$v_1 = \frac{-(1.5) + \sqrt{1.5^2 -4\times 1 \times (-20)}}{2(1)} \approx 3.78 $$ $$v_2 = \frac{-(1.5) - \sqrt{1.5^2 -4\times 1 \times (-20)}}{2(1)} \lt 0$$
The speed can't be negative.
$v_{run} = v = 3.78 \space mi/h$
Therefore, she runs at 3.78 mi/h