a) The perimeter is constrained by the exact amount of fencing available. Write an equation for this constraint based on the variable L and W. b) find a function that models the area of the rectangle as a function in terms of the width. c) find the largest possible area of the garden including the width that yields the maximum area. (use knowledge of the area being a quadratic function to describe your result algebraically)
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.
a) The perimeter is constrained by the exact amount of fencing available. Write an equation for this constraint based on the variable L and W.
b) find a function that models the area of the rectangle as a function in terms of the width.
c) find the largest possible area of the garden including the width that yields the maximum area. (use knowledge of the area being a quadratic function to describe your result algebraically)
Given:
The length of the fencing used to create 8 pens of the rectangular garden is 1200 feet and one side of the rectangular garden is already existing fence.
a)
Obtain the equation for the perimeter constrained based on the variable L and W, by the exact amount of fencing available.
The length of the 8 pens of the rectangular garden is 2L, excluding length of the existing fence.
The width of the 8 pens of the rectangular garden is 5W.
Therefore, the perimeter of the 8 pens in the rectangular garden is 2L+5W.
Here, the amount of fencing available is 1200 feet.
That is, the perimeter of the 8 pens of the ground is .
b)
Obtain the function that represents the area of the rectangle in terms of the width .
By (a), the perimeter of the 8 pens .
Rewrite the above equation L in terms of W.
Therefore, .
Compute the area of the rectangle A in terms of W.
Therefore, the function that represents the area of the rectangle .
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