a farmer wishes to create a rectangular pen to contain as much area as possible using 120 meters of fencing. determine the dimentions

To maximize the area of a rectangular pen with a fixed perimeter of 120 meters, we can use the following approach:

### Step 1: Set up the problem
Let:
- \(x\) be the length of the pen (in meters).
- \(y\) be the width of the pen (in meters).

The perimeter \(P\) of the rectangle is given by:
\[
P = 2x + 2y = 120
\]

### Step 2: Express one variable in terms of the other
Solve for \(y\) in terms of \(x\):
\[
2x + 2y = 120
\]
\[
x + y = 60
\]
\[
y = 60 - x
\]

### Step 3: Write the area function
The area \(A\) of the rectangle is given by:
\[
A = x \times y
\]
Substitute \(y = 60 - x\) into the area formula:
\[
A(x) = x(60 - x)
\]
\[
A(x) = 60x - x^2
\]

### Step 4: Find the maximum area
To maximize the area, we need to find the value of \(x\) that maximizes \(A(x)\). This requires taking the derivative of \(A(x)\) with respect to \(x\) and setting it equal to zero.

Differentiate \(A(x)\) with respect to \(x\):
\[
\frac{dA}{dx} = 60 - 2x
\]
Set the derivative equal to zero to find the critical point:
\[
60 - 2x = 0
\]
\[
2x = 60
\]
\[
x = 30
\]

### Step 5: Determine the corresponding width
Substitute \(x = 30\) back into the equation for \(y\):
\[
y = 60 - x = 60 - 30 = 30
\]

### Conclusion
The dimensions of the pen that maximize the area are:
\[
\boxed{30 \text{ meters} \times 30 \text{ meters}}
\]
This means the pen should be a square with each side measuring 30 meters.

Would you like more details or have any questions?

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Here are 5 related questions to explore:
1. Why does the area reach its maximum when the pen is a square?
2. How would the dimensions change if the perimeter were different, say 200 meters?
3. What if the farmer only had 90 meters of fencing—what would be the optimal dimensions?
4. How does the area function \(A(x) = 60x - x^2\) behave, and why?
5. Could the problem be solved using the method of Lagrange multipliers, and how?

**Tip:** When dealing with optimization problems involving perimeter and area, it's often helpful to express the area as a function of a single variable, then use calculus to find the maximum value.

Learn how to maximize the area of a rectangular pen using 120 meters of fencing with a step-by-step solution. This problem involves geometry and optimization techniques, focusing on perimeter and area calculations. Suitable for students in Grades 7-9.

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