Car A is traveling east at 60 mi/h and Car B is traveling north at 70 mi/h (see

Question

Car A is traveling east at 60 mi/h and Car B is traveling north at 70 mi/h (see the diagram below). At noon, Car A is 8 miles west of Detroit and Car B is 6 miles north of Detroit. At what rate is the distance between the two cars changing at this time? a) Identify the known rates, final values, and unknown rate (define your variables) b) Set up an equation relating the variables e) Use implicit differentiation d) Solve for the unknown rate and use this to determine whether a variable is increasing or decreasing

Explanation / Answer

Lets assume our coordinate system keeping detroit at origin where east shows positive X-axis and north shows positive Y-axis

Then known values are

a) x = -8 miles (as it is west of detroit)

y =6 miles (as it is north of detroit)

And dx/dt = 60 mil/hr (Directed towards east, so positive)

And dy/dt = 70 miles/hr (Directed towards north, so positive)

b) to set an equation relating their distance, use pythagorean theorem

s^2 = x^2 + y^2 (Always, irrespective of their speed , as they are moving perpendicular to each other)

c) Now we need to use implicit differentiation to find rate at which their distance are changing

So using

s^2 = x^2 + y^2

S for the time now is s^2 =8^2 + 6^2 =100

S =10 miles

And differentiating wrt

So 2s ds/dt = 2x dx/dt + 2y dy/dt

Putting values

2*10*dS/dt = 2*-8 *60 + 2*6* 70

dS/dt = -120/20 = -6 miles/hr

So solving for unknown , we find

Distance between A and B is decreasing at the rate of 6 miles/hr (ans)

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