Answer:
Danielle drove 3 hours
Step-by-step explanation:
Suppose Heather moved in the x direction.
Then Danielle moved in the opposite direction, that is, she moved in the -x direction.
Let's call [tex]V_1[/tex], [tex]x_1[/tex], [tex]t_1[/tex] at the speed, distance and time that I code Heather
Let's call [tex]V_2[/tex], [tex]x_2[/tex], [tex]t_2[/tex] at the speed, distance and time Danielle codes.
Observe the following diagram:
(x) Heather <------ [tex]x_1[/tex] --------- hospital ---------- [tex]x_2[/tex] -----------> Danielle (-x)
So:
[tex]V_1 = 35\ km/h[/tex]
[tex]t_1 = 4\ h[/tex]
[tex]V_2 = 50\ km/h[/tex]
We know that after 4 hours the distance between Heather and Danielle was 290 km.
That is to say:
[tex]x_2 + x_1 = 290\ km/h[/tex]
We know that [tex]x_1= v_1t_1[/tex]
[tex]x_1 = 35(4)[/tex]
[tex]x_1 = 140\ km[/tex]
We already know [tex]x_1[/tex], the distance from the hospital to Heather, so we can find [tex]x_2[/tex] and thus know [tex]t_2[/tex]
So:
[tex]x_2 = 290 - 140\\\\x_2 = 150\ km[/tex]
Then:
[tex]t_2 = \frac{x_2}{v_2}\\\\t_2 = \frac{150}{50}\\\\t_2 = 3\ h[/tex]
