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Question

 road perpendicular to a highway leads to a farmhouse located 5 mile away. An automobile traveling on the highway passes through this intersection at a speed of 50mph.

How fast is the distance between the automobile and the farmhouse increasing when the automobile is  miles past the intersection of the highway and the road?

The distance between the automobile and the farmhouse is increasing at a rate of  _____ miles per hour.

Answer 1

I believe you're missing a value in the question: How fast is the distance between the automobile and the farmhouse increasing when the automobile is___miles past the intersection of the highway and the road?

For this problem you'll want to use the Pythagorean theorem to set it up.

a2 + b2 = c2

One of the legs is 5 miles and is fixed (not changing), so it can be plugged in.

52 + b2 = c2

The other two distances b (distance between the intersection and the car) and c (distance between the car and the farmhouse), are changing with time. The question is asking what the rate of change of c (dc/dt) is.

Differentiating with respect to time yields

0 + 2b(db/dt) = 2c(dc/dt)

b is the number that's missing from the question. db/dt is 50. c can be solved for using the Pythagorean theorem with 5 and the value for b as the legs. Once you know all of those, you can plug them into the differentiated equation and solve for dc/dt.