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...

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

Asked on by utilityfan

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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.

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