To solve this problem, let’s go through it step-by-step.

Given:

• A rectangular field with three walls of fencing, with the fourth side against the gym wall.
• The area of the field is $360,000 \, \text{ft}^2$.
• The cost of fencing is $89 per foot.

Goal:

Minimize the fencing cost while maintaining the area at 360,000 ft².

Step 1: Define Variables

Let:

• $x$ = length of the side parallel to the gym wall (no fencing needed here).
• $y$ = width of the field, which will have fencing.

Step 2: Set Up the Area Equation

The area of the rectangle is given by: $x \cdot y = 360,000$ Thus, we can express $x$ in terms of $y$: $x = \frac{360,000}{y}$

Step 3: Set Up the Fencing Cost Function

Since only three sides need fencing (two widths $y$ and one length $x$), the total length of fencing required is: $\text{Total Fencing} = x + 2y$ Substitute $x$ from the area equation: $\text{Total Fencing} = \frac{360,000}{y} + 2y$

Step 4: Formulate the Cost Function

The cost $C$ in dollars is: $C = 89 \cdot \left( \frac{360,000}{y} + 2y \right)$ Expanding this: $C = 89 \cdot \frac{360,000}{y} + 178y$

Step 5: Differentiate the Cost Function with Respect to $y$ and Find Critical Points

To minimize $C$, we take the derivative of $C$ with respect to $y$ and set it equal to zero.

1. Differentiate $C$ with respect to $y$: $\frac{dC}{dy} = -\frac{89 \cdot 360,000}{y^2} + 178$

2. Set $\frac{dC}{dy} = 0$ to find the critical point: $-\frac{89 \cdot 360,000}{y^2} + 178 = 0$ Solving for $y$: $\frac{89 \cdot 360,000}{y^2} = 178$ $y^2 = \frac{89 \cdot 360,000}{178}$ $y^2 = 180,000$ $y = \sqrt{180,000} = 424.26 \, \text{ft}$

3. Substitute $y = 424.26$ back into the area equation to find $x$: $x = \frac{360,000}{424.26} \approx 848.53 \, \text{ft}$

Step 6: Calculate the Total Cost

Now that we have the dimensions:

• $x = 848.53$ ft
• $y = 424.26$ ft

The total fencing length needed is: $\text{Total Fencing} = x + 2y = 848.53 + 2 \cdot 424.26 = 1697.05 \, \text{ft}$

The cost is: $C = 89 \cdot 1697.05 \approx 151,438.45$

Final Answer:

The least amount of money needed is $151,438.45.

Would you like further details on any of these steps, or have any questions?

Related Questions:

1. What is the general method for minimizing costs in fencing problems?
2. How would the cost change if the price per foot of fencing increased?
3. What if the field area were different? How would that affect the dimensions?
4. Why do we only differentiate with respect to $y$ in this problem?
5. How can we verify that this solution gives the minimum cost?

Tip:

In optimization problems, always check if critical points provide a minimum or maximum by evaluating endpoints or using the second derivative test.