Shortest distance between two cars

[Adjoint]

Homework Statement

Two straight roads, which are perpendicular to each other, cross at point O.

Suppose a car is at distance 250m from the origin on one road, and another car is at distance 350m from the origin on another road.

Both cars are approaching towards the origin.

The first car has a constant velocity of 6m/s and the second car has constant velocity of 12m/s.

When does the distance between the two cars become shortest? And what's that shortest distance?

Homework Equations

The Attempt at a Solution

Lets suppose at time t the cars' distance becomes shortest.
So at that time the first car's position will be (0, 250 - 6t) and the second car's position would be (350 - 12t, 0)

So distance between them is √{(350 - 12t)² + (250 - 6t)²}

Next suppose A = (350 - 12t)² + (250 - 6t)²
For minimum dA/dt = 0 from here I get t

Is my approach ok? (I am not much expert in calculus.)

 

[Orodruin]

Looks reasonable. Solve for t in dA/dt = 0 and insert into your expression for the distance.

 

[Adjoint]

Thanks.

A = 180t² - 11400t + 185000
dA/dt = 360t - 11400 = 0 gives t = 31.7
And the minimum distance is √{(350 - 12t)² + (250 - 6t)²} = 67.1

 

[rude man]

Agreed.