RELATED RATE AND OPTIMIZATION PROBLEMS

Emily starts at point O at 8:00 am, and walks (5 km/h)to the East. Matthew starts at point B at 9:00 am, and walks (7 km/h) to the South. Question 1 (RELATED RATE PROBLEM): At what rate are Emily and Matthew separated at 11:00 am? Question 2:  (OPTIMIZATION PROBLEM) How many hours later Emily and Matthew will be closest?

QUESTION 1 (RELATED RATE PROBLEM):

At what rate are Emily and Matthew separated at 11:00 am?

Let's say:   t: The length of time that Emily has traveled

OA:  The distance that Emily has traveled at t hours:  x = 5t

BC:  The distance that Matthew has traveled at t hours:  7 (t - 1) = 7t -7

r² = OA² + OC²

r² = (5t)² + [20- (7t - 7)]²

r² = (5t)² + (27- 7t)²

r² = 25t² + (729- 378t + 49t²)

r² = 74t² - 378t + 729

Differentiate both sides of the equation with respect to t

r = (74t² - 378t + 729)^1/2

r = (74 x 3² - 378 x 3 + 729)^1/2

r = 16.16 km

Alternative solution to find the derivative

r = (74t² - 378t + 729)^1/2

QUESTION 2 (OPTIMIZATION PROBLEM):

How many hours later Emily and Matthew will be closest?

r = (74t² - 378t + 729)^1/2

148t - 378 = 0

t = 2.55

for t = 2.55

r = (74t² - 378t + 729)^1/2

r = (74 x 2.55² - 378 x 2.55 + 729)^1/2

r = 15.74 km (this is the closest distance they would have)