Use Derivatives to solve problems: Distancetime Optimization
A problem to minimize (optimization) the time taken to walk from one point to another is presented. First an applet is used to fully understand the problem and then an analytical method, using derivatives and other calculus concepts and theorems, is developed in order to find an analytical solution to the problem.
Problem : You decide to walk from point A (see figure below) to point C. To the south of the road through BC, the terrain is difficult and you can only walk at 3 km/hr. However, along the road BC you can walk at 5 km/hr. The distance from point A to the road is 5 km. The distance from B to C is 10 km. What path you have to follow in order to arrive at point C in the shortest ( minimum ) time possible?
Interactive Tutorial
We first try to understand the problem using the applet below. There are several possible paths one can follow to go from A to C. On the left panel of the applet, is shown possible paths: You may walk from point A to a certain point P, somewhere on the road between B and C, and continue along the road to get to point C. The question is: What is the position of point P that will minimize the time taken to go from A to C?
APPLET
Use the mousse to press and drag point P. What you are doing here is changing distance BP = x. On the right panel you have the time plotted against x. As you can see there seem to be one value of x for which the time is smallest (minimum). You may also plot the whole graph using the "on" and "off" buttons above it.
The total time t taken from A to C is calculated as follows:
t = distance AP / 3 km/hr + distance PC / 5 km/hr
Analytical Tutorial
We now look at a solution using derivatives and other calculus concepts. Let distance BP be equal to x. Let us find a formula for the distances AP and PC. Using Pythagorean theorm, we can write:
distance AP = sqrt(5^{ 2} + x^{ 2})
distance PC = 10  x
We now find time t_{ 1} to walk distance AP.(time = distance / speed).
t_{ 1} = distance AP / 3 = sqrt(5^{ 2} + x^{ 2}) / 3
Time t_{ 2} to walk distance PC is given by
t_{ 2} = distance PC / 5 = (10  x) / 5
The total time t is found by adding t_{ 1} and t_{ 2}.
t = sqrt(5^{ 2} + x^{ 2}) / 3 + (10  x) / 5
we might consider the domain of function t as being all values of x in the closed interval [0 , 10]. For values of x such that point P is to the left of B or to the right of c, time t will increase.
To find the value of x that gives t minimum, we need to find the first derivative dt/dx (t is a functions of x).
dt/dx = (x/3) / sqrt(5^{ 2} + x^{ 2})  1/5
If t has a minimum value, it happens at x such that dt/dx = 0.
(x/3) / sqrt(5^{ 2} + x^{ 2})  1/5 = 0
Solve the above for x. Rewrite the equation as follows.
5x = 3sqrt(5^{ 2} + x^{ 2})
Square both sides.
25x^{ 2} = 9(5^{ 2} + x^{ 2})
Group like terms and simplify
16x^{ 2} = 225
Solve for x (x >0 )
x = sqrt(225/16) = 3.75 km.
dt/dx has one zero. The table of sign of the first derivative dt/dx is shown below.
The first derivative dt/dx is negative for x < 3.75, equal to zero at x = 3.75 and positive for x >3.75. Also the values of t at x = 0 and x = 10 (the endpoints of the domain of t) are respectively 3.6 hrs and 3.7 hrs. The value of t at x = 3.75 is equal to 3.3 hrs and its is the smallest. The answer to our problem is that one has to walk to point P such BP = 3.75 km then procced along the road to C in order to get there in the shortest possible time.
Exercises
1  Solve the same problem as above but with the following values.
solution to the above exercise
x = 6.26 km (rounded to 2 decimal places).
More references on
calculus problems 

Online Step by Step Calculus Calculators and SolversNew !
Factor Quadratic Expressions  Step by Step CalculatorNew !
Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers

Interactive HTML5 Math Web Apps for Mobile LearningNew !

Free Online Graph Plotter for All Devices
Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: February 2015
Copyright © 2003  2015  All rights reserved