RELATED RATE AND OPTIMIZATION PROBLEMS

 

PROBLEM 1:

 

 

PROBLEM 2:

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)