Optimization | fencing problem | three sides maximize area of rectangle calculus

A mathematical modeling problem is covered in this example where we are fencing in a rectangular area that only requires three side lengths of fence. We draw a picture for the situation, set up a constraint equation using the given amount of fence (perimeter equation), then set up an objective function that we wish to minimize, the area function in this case. Then we solve the constraint equation for a variable and substitute into the objective function to create a function that only contains one variable. We then take its derivative using the power rule and find a critical number/value by setting the derivative equal to zero and solving. This gives us one side length. To get the other, we substitute back into the constraint equation. We then calculate the maximum area by substituting both variables into our area function. Finally, we double check to make sure that it will be a maximum by taking the second derivative and using the second 2nd derivative test to see that the function will be concave down, thus giving us a maximum.

[natio]

This was so helpful. Thank you so much!