Optimization - Distance - Maple Help

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Optimization: A Distance Example

Main Concept

An optimization problem involves finding the best solution from all feasible solutions. One is usually solving for the largest or smallest value of a function, such as the shortest distance or the largest volume.

A minimum or maximum of a continuous function over a range must occur either at one of the endpoints of the range, or at a point where the derivative of the function is 0 (and thus the tangent line is horizontal). These are called critical points.

Steps

 1 Identify what value is to be maximized or minimized.
 2 Define the constraints.
 3 Draw a sketch or a diagram of the problem.
 4 Identify the quantity that can be adjusted, called the variable, and give it a name, such as x.
 5 Write down a function expressing the value to be optimized in terms of x.
 6 Differentiate the equation with respect to x.
 7 Set the equation to 0 and solve for $x$.
 8 Check the value of the function at the end points.

Problem: Every morning Tom leaves his house, gets water from the river, and takes it to the farm. What is the shortest possible path that Tom has to walk?

Let x be the distance downstream from the house at the point where Tom gets water from the river.

Adjust the value of $x$ using the slider to find value that minimizes the distance traveled.

 x =

Numerical solution

  ,   Total distance $\mathbit{T}$: $T$ $=$   $=$ Calculate first derivative: $=$ Set it to 0: $0$ $=$ Solve for an x value that minimizes T: $0$ $=$  $=$  $=$  ${\left(30-x\right)}^{2}\left({x}^{2}+{10}^{2}\right)$ $=$   $=$  $=$  $=$ $0$  $-300\mathbf{}\left(x+30\right)\mathbf{}\left(x-10\right)$ $=$ Choose the positive root: $x$ $=$ $10$





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