A rectangular area of 28,000 ft2 is to be fenced off. Two types of fencing material are being considered: Metal fencing which costs $2 per ft and Wooden fencing which costs $5 per foot. You will be finding the dimensions (length and width) and the minimum total cost for fencing the area for three different scenarios. In each case you should find the exact value of both dimensions and an approximation to the nearest foot, as well as the total cost for fencing the area (to the nearest dollar). Find the minimum cost for fencing the rectangular area if metal fencing is used for three of the sides and wooden fencing is used for the fourth side. What would be the length and width of this rectangle (exact value and rounded to the nearest foot)?
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 rectangular area of 28,000 ft2
is to be fenced off. Two
types of fencing material are being considered: Metal fencing which
costs $2 per ft and Wooden fencing which costs $5 per foot. You will
be finding the dimensions (length and width) and the minimum total
cost for fencing the area for three different scenarios. In each case you
should find the exact value of both dimensions and an approximation
to the nearest foot, as well as the total cost for fencing the area (to the
nearest dollar).
Find the minimum cost for fencing the rectangular area if metal
fencing is used for three of the sides and wooden fencing is used
for the fourth side. What would be the length and width of this
rectangle (exact value and rounded to the nearest foot)?
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