A rectangular recreational field needs to be built outside of a gymnasium. Three walls of fencing are needed and the fourth wall is to be a wall of gymnasium itself. The ideal area for such a field is exactly 490000 ft^2. to minimize costs, it is necessary to construct the fencing using the least amount of material possible. Assuming the the material used in the fencing costs $41/ft, what is the least amount of money needed to build this fence of ideal area? 


To solve this problem, we need to minimize the cost of constructing a fence for a rectangular field of area 490,000 square feet, where only three sides require fencing. We’ll set up an optimization problem, where we minimize the length of fencing while achieving the required area.

### Step 1: Define Variables
1. Let $ x $ be the length of the side parallel to the gym wall (which won’t need fencing).
2. Let $ y $ be the width of the field, where both widths and one length will require fencing.

### Step 2: Set Up the Area Constraint
The area of the field is given by:
\[
x \times y = 490,000 \text{ ft}^2
\]
Thus:
\[
y = \frac{490,000}{x}
\]

### Step 3: Express the Fencing Requirement
The fencing is needed for three sides: two widths and one length. The total fencing required, $ F $, is:
\[
F = 2y + x
\]
Substitute $ y = \frac{490,000}{x} $:
\[
F = 2\left(\frac{490,000}{x}\right) + x = \frac{980,000}{x} + x
\]

### Step 4: Minimize the Fencing Requirement
To minimize $ F $, take the derivative with respect to $ x $, set it to zero, and solve for $ x $.

1. Differentiate:
   \[
   \frac{dF}{dx} = -\frac{980,000}{x^2} + 1
   \]

2. Set the derivative to zero:
   \[
   -\frac{980,000}{x^2} + 1 = 0
   \]
   \[
   \frac{980,000}{x^2} = 1
   \]
   \[
   x^2 = 980,000
   \]
   \[
   x = \sqrt{980,000} \approx 990 \text{ ft}
   \]

3. Substitute $ x = 990 $ into the area constraint to find $ y $:
   \[
   y = \frac{490,000}{990} \approx 495 \text{ ft}
   \]

### Step 5: Calculate the Fencing Cost
The total fencing required is:
\[
F = 2y + x = 2(495) + 990 = 1980 \text{ ft}
\]

Since the fencing costs $41 per foot:
\[
\text{Total Cost} = 1980 \times 41 = 81,180 \text{ dollars}
\]

### Final Answer
The least amount of money needed to build the fence is **$81,180**.

Would you like more details, or do you have any questions?

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#### Related Questions:
1. How would the solution change if the fencing cost were different?
2. What happens if the gymnasium wall is not parallel to the field length?
3. Could this approach be adapted for a different target area?
4. How does the minimum fencing requirement change if the area requirement changes?
5. What if the field is square instead of rectangular?

#### Tip:
In optimization problems involving geometry, drawing a diagram can help clarify which variables need to be minimized or maximized.

A rectangular recreational field needs to be built outside of a gymnasium. Three walls of fencing are needed and the fourth wall is to be a wall of gymnasium itself. The ideal area for such a field is exactly 490000 ft^2. To minimize costs, it is necessary to construct the fencing using the least amount of material possible. Assuming the material used in the fencing costs $41/ft, what is the least amount of money needed to build this fence of ideal area?

Find the least cost to build a fence for a rectangular field of 490000 ft² using only three sides of fencing. Learn about optimization and cost minimization in geometry with calculus. Suitable for high school students.

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