Question

Heather left the hospital and drove toward the desert at an average speed of 35 km/h. Danielle left sometime later driving in the opposite direction with an average speed of 50 km/h. After Heather had driven for four hours they were 290 km apart. How long did Danielle drive

Answer 1

Danielle drove 3 hours

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
`V_1`,
`x_1`,
`t_1` at the speed, distance and time that I code Heather

Let's call
`V_2`,
`x_2`,
`t_2` at the speed, distance and time Danielle codes.

Observe the following diagram:

(x) Heather <------
`x_1` --------- hospital ----------
`x_2` -----------> Danielle (-x)

So:

`V_1 = 35\ km/h`

`t_1 = 4\ h`

`V_2 = 50\ km/h`

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`

We know that
`x_1= v_1t_1`

`x_1 = 35(4)`

`x_1 = 140\ km`

We already know
`x_1`, the distance from the hospital to Heather, so we can find
`x_2` and thus know
`t_2`

So:

`x_2 = 290 - 140\\\\x_2 = 150\ km`

Then:

`t_2 = (x_2)/(v_2)\\\\t_2 = (150)/(50)\\\\t_2 = 3\ h`