Problem 16 - Contemporary Precalculus

Question

A fence is needed to enclose an area of 30,246 square feet. One side of the area is bounded by an existing fence, so no new fencing is needed there. Fencing for the side opposite the existing fence costs \(\$ 18\) per foot. Fencing for the other two sides costs \(\$ 6\) per foot. What is the cost of the least expensive fence?

Step-by-Step Solution

Answer

Answer: The minimum cost to build the fence is approximately $6,011.37. The dimensions of the fence to achieve this cost are approximately 100.82 feet for the side with cheaper fencing and 300.16 feet for the other side with more expensive fencing.

Step 1: Define the Variables

Let \(x\) be the length of the side with the cheaper fencing (\(\$6\) per foot) and \(y\) be the length of the side with the more expensive fencing (\(\$18\) per foot). Since one side of the fence is already built, we are enclosing a rectangular area with fencing on three sides.

Step 2: Write the Area Equation

Given that the area enclosed is 30,246 square feet, we can write the area equation as: \[xy = 30246\]

Step 3: Write the Cost Function

The cost function, \(C(x)\), can be written based on the cost of fencing for each side. Since the cheaper fencing costs \(\$ 6\) per foot, and there are two sides of length \(x\), the cost for the cheaper fencing is \(6x + 6x = 12x\). The more expensive fencing costs \(\$18\) per foot, and the length of the side is \(y\). The total cost function is: \[C(x) = 12x + 18y\]

Step 4: Express y in terms of x using the Area Equation

To minimize the cost, we need to express \(y\) in terms of \(x\) using the area equation. From the equation \(xy = 30246\), we isolate \(y\): \[y=\frac{30246}{x}\]

Step 5: Substitute y into the Cost Function

Now, we substitute this expression for \(y\) into the cost function \(C(x)\): \[C(x) = 12x + 18\left(\frac{30246}{x}\right)\]

Step 6: Find the x value that minimizes the cost function

To find the minimum cost, we need to find the derivative of the cost function and set it equal to 0, then solve for \(x\). Calculate the derivative of \(C(x)\) with respect to x: \[C'(x) = 12 - \frac{18 \cdot 30246}{x^2}\] Now, set the derivative equal to 0 and solve for x: \[0 = 12 - \frac{18 \cdot 30246}{x^2}\] \[x^2 = \frac{18 \cdot 30246}{12}\] \[x = \sqrt{\frac{18 \cdot 30246}{12}} \approx 100.82\]

Step 7: Find the fencing dimensions

Now that we have the value of \(x\), we can find the corresponding value of \(y\) using the expression we found in Step 4: \[y=\frac{30246}{100.82} \approx 300.16\]

Step 8: Calculate the Minimum Cost

Finally, we can calculate the minimum cost using the cost function from Step 3 and plugging in our dimensions for \(x\) and \(y\): \[C(100.82) = 12(100.82) + 18(300.16) \approx \$6011.37\] The least expensive fence will cost approximately \(\$6011.37\).

Key Concepts

Cost Function

In optimization problems, the cost function represents the total cost of the elements involved in a particular situation. Here, we are dealing with constructing a fence to enclose a given area, where different lengths of the fence come at different costs.

For this problem, the cost function is calculated considering two types of fencing costs: one at \(6 per foot for two sides, and the other at \)18 per foot for one side. The task is to express the total cost of the three sides we're constructing in terms of a single variable. This results in the cost function:

• Two sides of fence each with length x and cost \(6 per foot: total cost = \)12x.
• One side of fence with length y and cost \(18 per foot: total cost = \)18y.

The total cost function becomes:\[ C(x) = 12x + 18y \]

This function will allow us to calculate the total cost of the fencing once we know the dimensions of the enclosure.

Area Equation

An area equation is used to relate the dimensions of a geometric shape to its area. In this scenario, we are dealing with a rectangular area that needs to be fenced on three sides.

Given:

• The total area to be enclosed is 30,246 square feet.

With one of the sides already fenced, we establish the area equation as:\[ xy = 30246 \]

Here, \( x \) is the length of the two sides with cheaper fencing, and \( y \) is the length of the side with the more expensive fencing. Rearranging this equation helps in simplifying the cost function by expressing \( y \) in terms of \( x \), which is crucial for minimizing the total cost.

Derivative

In calculus, a derivative is used to determine the rate of change of a function. When optimizing, we use derivatives to find local minima or maxima of the function.

For minimizing our cost function, we first need the derivative of the cost function \( C(x) = 12x + 18y \) (where \( y \) is expressed in terms of \( x \) as shown in the area equation):\[ C(x) = 12x + 18\left(\frac{30246}{x}\right) \]

To find the optimal value of \( x \) that minimizes cost, we take the derivative of \( C(x) \) with respect to \( x \):\[ C'(x) = 12 - \frac{18 \cdot 30246}{x^2} \]

Setting the derivative \( C'(x) \) to zero and solving for \( x \) allows us to find the critical point, which tells us where the function reaches its minimum cost point.

Minimization

The goal of minimization is to find the point at which a function reaches its lowest possible value. In this fencing problem, we aim to minimize the total cost of the fence.

Minimization involves:

• Expressing all variables in the cost function in terms of a single variable using the area equation.
• Finding the derivative of the cost function, \( C'(x) \), and setting it to zero to find critical points.
• Solving for the value of \( x \) that achieves this minimum cost.

After calculating, we find \( x \approx 100.82 \).

Using this \( x \) value in the area equation allows y to be calculated, ensuring the area requirement is met, and verifying that this configuration indeed provides the minimum cost for constructing the fence. Concluding, the calculation shows the least expensive fence configuration costs approximately $6011.37.