How fast are the cars separating 2 minutes after car B passes through the intersection in the following case:Highways 29 and 90 meet at right angles at Sioux Falls, S.D.  Car A traveling south on...

How fast are the cars separating 2 minutes after car B passes through the intersection in the following case:

Highways 29 and 90 meet at right angles at Sioux Falls, S.D.  Car A traveling south on route 29 at 60 mph passes through the intersection 3 minutes before car B traveling east on route 90 at 45 mph.  How fast are the cars separating 2 minutes after car B passes through the intersection?

Asked on by paulr162

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Highways 29 and 90 meet at right angles at Sioux Falls, S.D.  Car A traveling south on route 29 at 60 mph passes through the intersection 3 minutes before car B traveling east on route 90 at 45 mph.  

At the time two minutes after car B passes through the intersection, car A has travel for 5 minutes and car B has traveled for 2 minutes.

The distance traveled by A is 60*5/60 = 5 miles and the distance traveled by B is 45*2/60 = 1.5 miles.

The distance between the cars is D^2 = x^2 + y^2

Using implicit differentiation:

`2*D*(dD)/(dt) = 2x*(dx/dt) + 2*y*(dy/dt)`

At the given moment two minutes after B crosses the intersection

`2*sqrt(5^2 + 1.5^2)*(dD)/(dt) = 10*60 + 3*45`

=> `(dD)/(dt) = (10*60 + 3*45)/(2*sqrt(5^2 + 1.5^2))`

=> `(dD)/(dt) = 735/(2*sqrt(27.25))`

The rate at which the two cars are moving apart is `735/(2*sqrt(27.25))` `~~` 70.4 mph

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