Solve the following problems involving optimization. 1. A rectangular field is to be fenced off along the bank river where no fence is required along the bank. If the material for the fence costs 60 pesos per running foot for the two ends and 90 pesos per running foot for the side parallel to the river, find the dimensions of the field of largest possible area that can be enclosed with 27,000 pesos worth of fence.
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.
Solve the following problems involving optimization.
1. A rectangular field is to be fenced off along the bank river where no
fence is required along the bank. If the material for the fence costs 60
pesos per running foot for the two ends and 90 pesos per running foot
for the side parallel to the river, find the dimensions of the field of largest possible area that can be enclosed with 27,000 pesos worth of fence.
2. Find a pair of non-negative number that have a product of 162 and
minimize the sum of two times the first number and second number
with closed bounded interval of [1 10]
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