Lightning McQueen is traveling east at 80 km/h. At an intersection 50 km ahead,

Question

Lightning McQueen is traveling east at 80 km/h. At an intersection 50 km ahead, Mack is traveling north at 50 km/h. How far apart will the vehicles be when they are closest to each other?
Lightning McQueen is traveling east at 80 km/h. At an intersection 50 km ahead, Mack is traveling north at 50 km/h. How far apart will the vehicles be when they are closest to each other?
Lightning McQueen is traveling east at 80 km/h. At an intersection 50 km ahead, Mack is traveling north at 50 km/h. How far apart will the vehicles be when they are closest to each other?

Explanation / Answer

Let us denote the Lightning McQueen by Q and  Mack by M.

Q's speed = vq = 80 km/h

M's speed = vm = 50 km/h

Now, initially M is 50 km east of Q. Supose they are closest at time t

In this time distance travelled by Q is L = vqt = 80t

In this time distance travelled by M is D = vmt = 50t

distance between them at time t0 is S = [(50-L)2 + D2] as shown in the figure:

or S = [50 - 80t)2 + (50t)2] ------------ (1)

differentiating on both sides with respect to time we get

or dS/dt = [-160(50-80t) + 100t]/[2[50 - 80t)2 + (50t)2]]

if they are closest at time t then S would be minimum and thus dS/dt = 0 which gives

or [-160(50-80t) + 100t] = 0

or t = 400/645hrs

So the closest distance is given by putting this value of t in equation (1) it gives:

S = [50 - 80t)2 + (50t)2] = 31km

So, they will be 31km apart when they are closest to each other.

This concludes the answers. If you find anything lacking please let me know.. I will resolve your query without delay....

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