A road perpendicular to a highway leads to a farmhouse located 10 mile away. An automobile traveling on the highway passes through this intersection at a speed of 70mph.

Question

A road perpendicular to a highway leads to a farmhouse located 10 mile away. An automobile traveling on the highway passes through this intersection at a speed of 70mph.

How fast is the distance between the automobile and the farmhouse increasing when the automobile is 9 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

| 10 miles

+------------------Farm--->x

|

|

| dy/dt = 70 mph

|

y

car

Distance, s(t) between the car and the farm is the hypotenuse to the right triangle.

s² = 10² + y²

Take the derivative of both sides of the equation

2sds/dt = 2ydy/dt

Need to solve for ds/dt

ds/dt = y/s dydt = (y/s) (70)

When y = 9.

s = √(100+81) = 13.45 miles

ds/dt = (9/13.45)(70) = 46.84 mph