A problem in minima and maxima applicatios ( )please

  • Thread starter Thread starter faisal-tesla
  • Start date Start date
  • Tags Tags
    Maxima Minima
Click For Summary
SUMMARY

The discussion centers on optimizing the cost of laying a cable across a river and along its bank. The problem involves determining the optimal point C on the north bank to minimize costs, given that the cable costs M per kilometer in water and N per kilometer along the bank, with M being greater than N. The solution requires evaluating the function at critical points, specifically at x=0, x=9, and the point where the derivative equals zero, to confirm the minimum cost location.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and critical points.
  • Knowledge of optimization techniques in mathematical problems.
  • Familiarity with cost functions and their applications in real-world scenarios.
  • Basic concepts of geometry related to straight lines and distances.
NEXT STEPS
  • Study optimization problems involving piecewise functions.
  • Learn about differentiating functions to find minima and maxima.
  • Explore real-world applications of calculus in cost minimization.
  • Investigate the implications of varying cost parameters M and N on the solution.
USEFUL FOR

Students in mathematics, particularly those studying calculus and optimization, as well as professionals in fields requiring cost analysis and resource allocation strategies.

faisal-tesla
Messages
4
Reaction score
0
a problem in minima and maxima applicatios (urgent !)please

Homework Statement



A straight river 3km wide runs due east. A cable is to be laid from a point A on the south bank of the river to a point B on its north bank, 9km downstream from A. The cable is to be laid in a straight line from A to some point C on the north bank, and from there it is to be laid in a straight line along the north bank to B. If cable laid through water costs $M per kilometre and cable laid along the bank costs $N per kilometre with M > N, where should C be located to minimise the total cost of the cable? What is the minimum cost? Prove that your answer corresponds to the minimum cost


Homework Equations





The Attempt at a Solution



I got a number in terms of N and M. I also have to check the end point (x=0, x=9)

However, I couldn't prove the value is a minimum because it basically depends on N/M


regards,
Faisal
 
Physics news on Phys.org


If the function is differentiable and you have exactly one point x in [0,9] where the derivative is zero, then the only candidates for the locations of the minimum and maximum are x, 0, and 9. So just check the function value at those three points: whichever one is smallest is the minimum.
 

Similar threads

How Much Wire Should Be Laid Underwater to Minimize Cost?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Finding the maxima and minima of a surface defined by datapoints
  • · Replies 5 ·
Replies
5
Views
3K
Finding the derivative in an optimization problem
  • · Replies 8 ·
Replies
8
Views
4K
Optimizing Wire Usage for Telephone Service Installation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Solving the Boat Crossing a River Problem using the Calculus of Variations
  • · Replies 33 ·
2
Replies
33
Views
5K
MHB Jesus' question at Yahoo Answers regarding finding total cost function
  • · Replies 1 ·
Replies
1
Views
2K
How Do You Calculate the Minimum Cost of Constructing a Power Line?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How Should Cable Be Laid to Minimize Costs Across a River?
  • · Replies 2 ·
Replies
2
Views
4K
The minimum cost of a power line
  • · Replies 1 ·
Replies
1
Views
3K
How Do You Calculate a Boat's Path Across a River with a Current?
  • · Replies 1 ·
Replies
1
Views
3K