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
,
,
at the speed, distance and time that I code Heather
Let's call
,
,
at the speed, distance and time Danielle codes.
Observe the following diagram:
(x) Heather <------
--------- hospital ----------
-----------> Danielle (-x)
So:
![t_1 = 4\ h](https://tex.z-dn.net/?f=t_1%20%3D%204%5C%20h)
![V_2 = 50\ km/h](https://tex.z-dn.net/?f=V_2%20%3D%2050%5C%20km%2Fh)
We know that after 4 hours the distance between Heather and Danielle was 290 km.
That is to say:
![x_2 + x_1 = 290\ km/h](https://tex.z-dn.net/?f=x_2%20%2B%20x_1%20%3D%20290%5C%20km%2Fh)
We know that ![x_1= v_1t_1](https://tex.z-dn.net/?f=x_1%3D%20v_1t_1)
![x_1 = 35(4)](https://tex.z-dn.net/?f=x_1%20%3D%2035%284%29)
![x_1 = 140\ km](https://tex.z-dn.net/?f=x_1%20%3D%20140%5C%20km)
We already know
, the distance from the hospital to Heather, so we can find
and thus know ![t_2](https://tex.z-dn.net/?f=t_2)
So:
![x_2 = 290 - 140\\\\x_2 = 150\ km](https://tex.z-dn.net/?f=x_2%20%3D%20290%20-%20140%5C%5C%5C%5Cx_2%20%3D%20150%5C%20km)
Then:
![t_2 = \frac{x_2}{v_2}\\\\t_2 = \frac{150}{50}\\\\t_2 = 3\ h](https://tex.z-dn.net/?f=t_2%20%3D%20%5Cfrac%7Bx_2%7D%7Bv_2%7D%5C%5C%5C%5Ct_2%20%3D%20%5Cfrac%7B150%7D%7B50%7D%5C%5C%5C%5Ct_2%20%3D%203%5C%20h)