Math Problem Statement
A farmer wants to build a rectangular pen along a straight river. She wants to divide the pen into 5 equal rectangular pieces as shown in the picture. She will not fence one of the long sides along the river. (a) What is the largest area she can enclose with 3,000 feet of fencing? (b) What are the length and width of the individual pens that will give the largest area?
Solution
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Math Problem Analysis
Mathematical Concepts
Optimization
Algebra
Differentiation
Quadratic Functions
Formulas
Fencing constraint: L + 6W = 3000
Area function: A = L * W = (3000 - 6W) * W = 3000W - 6W^2
Derivative: dA/dW = 3000 - 12W
Theorems
Maximization using first derivative
Quadratic Optimization
Suitable Grade Level
Grades 11-12 (Advanced High School)
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