Math Problem Statement
Solution
The problem you're solving involves finding the dimensions of five identical pens arranged side by side, where the total area is , and we need to minimize the amount of fencing used. Here's how we approach it:
1. Area Equation:
The total area of the five pens is given by: where is the length of the pens and is the width of each pen. This equation can be rewritten to solve for :
2. Fence Equation:
The total length of fencing required (perimeter) is given by: This includes the lengths of the horizontal sections (6 , as there are six vertical boundaries) and the vertical sections (10 , as there are five pens requiring five boundaries and two external ones).
3. Substituting into the Fence Equation:
Now substitute into the equation for the fence:
4. Minimizing the Fence Equation:
To minimize the amount of fence used, take the derivative of with respect to and set it equal to zero: Setting this equal to zero:
5. Finding :
Now, substitute back into the equation for :
Final Dimensions:
- Width
- Length
These dimensions will minimize the amount of fencing used while keeping the total area of the pens at .
Would you like more details or have any questions?
Related Questions:
- How do we confirm that the calculated dimensions give the correct total area of ?
- Can we use Lagrange multipliers to solve this optimization problem?
- What happens to the fence equation if the number of pens changes from 5 to another value?
- What if the area constraint changes to something other than ?
- How does the shape of the pens (square vs rectangular) affect the fencing requirements?
Tip:
When optimizing a quantity, always check if the second derivative confirms a minimum or maximum!
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Math Problem Analysis
Mathematical Concepts
Optimization
Calculus
Geometry
Formulas
Area: 1000 = 5xy
Fence Equation: F = 6x + 10y
Minimization of F: dF/dy = -1200/y^2 + 10
Theorems
Optimization using derivatives
Suitable Grade Level
Grades 11-12, College level
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