Suppose that you are the power company and must provide power to a lighthouse on a rock one mile off shore. The nearest pole to run the line from is 2 miles up the shore from the line straight out to the lighthouse. If the line costs $1000/mile on land and $1500/mile on water, find the path for the line that minimizes the total cost. Your answer should state how far from the pole your line leaves the shore and goes to the lighthouse. Show that this is a minimum and find the cost for the path you have chosen.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
Suppose that you are the power company and must provide power to a lighthouse on a rock
one mile off shore. The nearest pole to run the line from is 2 miles up the shore from the line
straight out to the lighthouse. If the line costs $1000/mile on land and $1500/mile on water, find
the path for the line that minimizes the total cost. Your answer should state how far from the
pole your line leaves the shore and goes to the lighthouse. Show that this is a minimum and find
the cost for the path you have chosen.
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