When will the two vehicles be closest together in the following case:
Two cars are traveling in different directions but their positions are given in reference to the intersection of Oxford and Highbury.
Car 1 is traveling east at 50 km/h and is 1 km east of the intersection.
Car 2 is traveling north at 70 km/h and is 5 km south of the intersection
This information was taken a 8:00 AM
At 8:00 AM, Car 1 is 1 km east of the intersection and is traveling east at 50 km/h and Car 2 is 5 km south of the intersection and is traveling north at 70 km/h.
Let the time when they are the closest together be t minutes after 8:00 AM. After t minutes Car 1 is 1+50*t/60 km from the intersection towards the East. After t minutes Car 2 is 5 - 70*t/60 km from the intersection towards the North. The distance between the cars can be determined using the Pythagorean Theorem as `D = sqrt((1+(5*t)/6)^2+(5-(7*t)/6)^2)`
=> D = `(1/6)*sqrt(36 + 25t^2 + 60t + 900 + 49t^2 - 420t)`
=> D = `(1/6)sqrt(74t^2 -360t + 966)`
Taking the derivative of D with respect to t gives:
`(dD)/dt = (1/6)(1/2)*(148t - 360)/sqrt(74t^2 - 360t + 966)`
When the cars are closest together `(dD)/dt = 0`
=> 148t - 360 = 0
=> t = 360/148
=> t = 2.43
The two cars are closest to each other at 2.34 minutes after 8:00 AM.