A farmer decides to enclose a rectangular field using the side of a barn as one side of the rectangle. The figure below shows the fenced area he wants to make. If he has 3100 feet of fencing, what values of x and y will maximize the enclosed area? x= ?feet y=?feet
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.
1-A farmer decides to enclose a rectangular field using the side of a barn as one side of the rectangle. The figure below shows the fenced area he wants to make.
If he has 3100 feet of fencing, what values of x and y will maximize the enclosed area?
x= ?feet
y=?feet
2-Suppose you have 1640 meters of fencing available with which to build three adjacent rectangular corrals as shown in the figure. Find the dimensions so that the total enclosed area is as large as possible.
smaller side =?
longer side equals?
3-Micah wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Micah has 800 feet of fencing, what dimensions would maximize the area of the pen?
a) Let w be the length of the pen perpendicular to the barn. Write an equation to model the area of the pen in terms of w
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