Problem 65 - College Algebra

Question

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Step-by-Step Solution

Answer

The dimensions that maximize the area are 150 feet by 300 feet, resulting in a maximum area of 45000 square feet.

Step 1: Define variables

Let's denote the width of the plot by \(x\) (this will be the side parallel to the river). The only length of fence will then be the length of the lot, denoted by \(y\). The total amount of fencing available is 600 feet, and since we're using it on two sides, we'll have the equation \(2x + y = 600\). Simplifying this, we get \(y = 600 - 2x\). We will use this in our next step.

Step 2: Define the function to be optimized

The area which we aim to maximize, for a rectangle, is given by product of its length and width, therefore, we have \(Area = x * y\). Replace \(y\) from the equation obtained in Step 1, and we get \(Area = x * (600 - 2x) = 600x - 2x^2\). This is the function we have to optimize.

Step 3 : Find the derivative and solve for x

To find the maximum area, take the derivative of the area function with respect to \(x\) and set it equal to zero. \(dArea/dx = 600 - 4x\). Setting this equal to zero, we find \(x = 150\).

Step 4: Compute length

Substitute \(x = 150\) back into the equation for \(y\) to calculate the length. We get \(y = 600 - 2*150 = 300\).

Step 5: Validate solution

Double check the solution \(x = 150\) by using the second derivative test. The second derivative of \(Area = -4\) which is less than zero, confirming that this will indeed give a maximum area.

Step 6: Compute Maximum Area

Finally, plug these dimensions back into the area function to get the maximum area: \(Area = 150*300 = 45000 square feet\).