How do I solve Calculus Optimization Questions?

Question

Person A leaves a house travelling 40 km/h north. Person B leaves the same house travelling 45 km/h east. If Person B begins 1 hour AFTER person A, at what rate are their cars seperating three hours after Person A begins?
I don't understand how to solve this question because of the time differences in their trips. I would appreciate any suggestions. Thanks!!

Answer 1

This is a vector problem - you have a person going one way and a person going another way, and you have to find the distance between them. Lucky for us, they are going at right angles to each other, so the distance between them will be found using the Pythagorean Theorem.

The first person is travelling 40 km/h north, so his position would be A = 40t in the y direction. The second person is going 45 km/h east, so his position would be B = 45(t - 1) in the x direction. The t - 1 term sets our time reference to person A - at 1 hour (or t = 1), the rate of separation is only controlled by A = 40t.

The rate of separation after 1 hour is our hypotenuse:

RoS = sqrt[(40t)^2 + [45(t - 1)]^2]
RoS = sqrt[1600t^2 + 2025(t^2 - 2t + 1)]
RoS = sqrt[1600t^2 + 2025t^2 - 4050t + 2025)]
RoS = sqrt(3625t^2- 4050t + 2025)
RoS = 5sqrt(145t^2- 162t + 81)

We can find the RoS after 3 hours - just plug it in and crunch the numbers:

RoS = 5sqrt(145(3)^2- 162(3) + 81)
RoS = 5sqrt(1305 - 486 + 81)
RoS = 5sqrt(900)
RoS = 5(30)
RoS = 150 km/h