Question

A four-wheel-drive vehicle is transporting an injured hiker to the hospital from a point that is 30 km from the nearest point on a straight road. The hospital is 70 km down that road from that nearest point. If the vehicle can drive at 30 kph over the terrain and at 130 kph on the road, how far down the road should the vehicle aim to reach the road to minimize the time it takes to reach the hospital? (Round your answer to two decimal places.)

Answer 1

Answer:

$D=200\ km$

Explanation:

distance on terrain, $d_t=30\ km$

• distance on the road, $d_r=70\ km$

• speed on terrain, $v_t=30\ km.hr^{-1}$

• speed  on road, $v_r=130\ km.hr^{-1}$

time taken on the terrain,

$t_t=\frac{d_t}{v_t}$

$t_t=\frac{30}{30}$

$t_t=1\ hr$

time taken to cover the distance on the road:

$t_r=\frac{d_r}{v_r}$

$t_r=\frac{70}{130}$

$t_r=\frac{7}{13}\ hr$

Now the distance covered on terrain in the total time:

$D= v_r\times (t_r+t_t)$

$D= 130\times (\frac{7}{13}+1)$

$D=130\times \frac{20}{13}$

$D=200\ km$

is the distance the vehicle must target on the road to minimize the time taken in going off the road.