Answer: 31 km
Step-by-step explanation:
Time, t = Distance, d/Velocity, v
Let's assume the ambulance drives a distance of x along the road, which leaves us 50-x on the road, after which it has reached the road.
It forms a triangle, after which we can use Pythagoras Theorem.
Using it, we get distance down the hill, d1, and we can calculate distance on the road remaining till the hospital, d2.
t = d1/v1 + d2/v2
= [tex]\frac{\sqrt{x^{2} + 30^{2} } }{30} + \frac{50 - x}{120}[/tex]
We need to minimise t. Therefore, we have to differentiate.
On differentiating anf equating it to 0, we get the value of x as [tex]\sqrt{60} = 7.75[/tex]
How far down the road should the vehicle aim to reach the road to reach the hospital at the minimum time?
[tex]= \sqrt{900 + 7.75^{2} } = 31 km[/tex]
