Related Rates Farmhouse question

Question

A road perpendicular to a highway leads to a farmhouse located 3 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 1 miles past the intersection of the highway and the road?

Answer 1

construct a right triangle, hypotenuse c, legs a and b

a=3 miles

b=1 mile

c=sqr(3^2+1^2)= sqr10 miles

b'=50 mph

a'=0

c'=5sqr10= about 15.81 mph

c^2=a^2+b^2

take the derivative, then divide by 2

cc'=aa'+bb'

c'sqr10=3(0)+ (1)50

c'=50/sqr10=50sqr10/10= 5sqr10 mph = rate of change in distance to the farmhouse

= about 15.81 mph