Calc 1 question

Question

A police car is located 60 feet to the side of a straight road.

A red car is driving along the road in the direction of the police car and is 140 feet up the road from the location of the police car. The police radar reads that the distance between the police car and the red car is decreasing at a rate of 100 feet per second. How fast is the red car actually traveling along the road?

The actual speed (along the road) of the red car is   feet per second

Answer 1

construct a right triangle with hypotenuse = distance from the car to the police = h

h^2 = a^2 + b^2 where a=60, b= 140, h' =100, a' = 0, b' = speed of the car

take the derivative (this is a "related rates" problem, usually a subtopic in the section of a calculus texbook titled "derivatives")

2hh' = 2aa' + 2bb'

divide by 2

hh' = aa' + bb'

plug in the values

100sqr(140^2 + 60^2) =60(0) + 140b'

solve for b'

b' = (100/140)sqr(140^2 + 60)^2

b' = (5/7)sqr(19600 +3600)

b' = (5/7)10sqr232

b' = about (50/7)15.23 ft/sec

b' = about 108.80 feet per second