The owner of a lumber store wants to construct a fence to enclose a rectangular outdoor storage area adjacent to the store, using part of the side of the store (which is 210 feet long) for part of one of the sides. (See the figure below.) There are 400 feet of fencing available to complete the job. Find the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure. FENCE? PARALLEL SIDE PERPENDICUL AR SIDE Length of parallel side(s) STORE Length of perpendicular sides =
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.
#### Diagram Explanation
The diagram depicts a store with a rectangular fenced area adjacent to it. The fencing is applied to:
- Part of the side of the store (210 feet long).
- The parallel and perpendicular sides of the outdoor storage area.
The fence is shown with dashed lines representing the three sides (one parallel to the store and two perpendicular sides forming the rectangular enclosure).
#### Calculation Inputs
To solve for the optimal lengths of the sides:
- Length of parallel side(s) = [Input Box]
- Length of perpendicular sides = [Input Box]
### Objective
Find the length of the sides parallel and perpendicular to the store that will maximize the total enclosed area using the available 400 feet of fencing.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c9fe6be-99fb-4863-a04f-7bda979173e6%2F788ff065-ce46-4eed-ba78-6c7ba7d60ea4%2F13wo2fg_processed.png&w=3840&q=75)
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