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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.16.b
Chapter 4, Problem 4.5.16.b

Pen problems


b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? <IMAGE>

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Define the variables: Let the width of each pen be 'w' meters and the length be 'l' meters. Since there are four pens, the total length along the barn is '4w'.
Set up the area equation: Each pen has an area of 100 m², so the equation for the area of one pen is 'w * l = 100'.
Express 'l' in terms of 'w': From the area equation, solve for 'l' to get 'l = 100 / w'.
Set up the perimeter equation: The total amount of fencing needed is the sum of the lengths of the sides not against the barn. This is '3l + 5w' (3 lengths and 5 widths).
Substitute 'l' in the perimeter equation: Replace 'l' with '100 / w' in the perimeter equation to get '3(100 / w) + 5w'. Simplify this expression to find the function of 'w' that represents the total fencing needed. Then, find the derivative and solve for 'w' to minimize the fencing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, the rancher seeks to minimize the total length of the fence while maintaining a fixed area for each pen. This requires setting up a function that represents the total fencing needed and using techniques such as derivatives to find critical points that yield the minimum value.
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Intro to Applied Optimization: Maximizing Area

Area and Perimeter Formulas

Understanding the formulas for area and perimeter is crucial in this problem. The area of each rectangular pen is given as 100 m², which can be expressed as length times width (l * w). The perimeter, or the amount of fencing needed, is determined by the dimensions of the pens, and the relationship between these dimensions must be established to solve the optimization problem effectively.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint

Constraints

Constraints are conditions that must be satisfied in optimization problems. In this case, the area of each pen is a constraint that limits the possible dimensions of the pens. By incorporating this constraint into the optimization process, we can express one variable in terms of another, allowing us to reduce the number of variables and simplify the problem to find the optimal dimensions.
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