Optimization: A Distance Example
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.
Identify what value is to be maximized or minimized.
Define the constraints.
Draw a sketch or a diagram of the problem.
Identify the quantity that can be adjusted, called the variable, and give it a name, such as x.
Write down a function expressing the value to be optimized in terms of x.
Differentiate the equation with respect to x.
Set the equation to 0 and solve for x.
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.
d = x2 +102 , D= 30−x2 + 202
x = 0≤x ≤30, x ∈ℝ
Total distance T:
x2+102 + 30−x2+202
Calculate first derivative:
Set it to 0:
Solve for an x value that minimizes T:
x 30−x2+202− 30−xx2+102
x2 − 60 x + 900x2+100
x2 x2 − 60 x + 900+400
x4 − 60 x3 + 1300 x2
−300⁢x2−6000 x +90000
Choose the positive root:
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