The owner of a lumber store wants to construct a fence to enclose a rectangular outdoor storage area adjacent to the store, using part of the side of the store (which is 240 feet long) for part of one of the sides. (See the figure below.) There are 430 feet of fencing available to complete the job. Find the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure. Note: you can click on the image to get a enlarged view. Length of parallel side(s) = Length of perpendicular sides $=$
Solution
Step 1 ：Let's denote the length of the side parallel to the store as \(x\) and the length of the side perpendicular to the store as \(y\). We know that the total length of the fencing is 430 feet, so we have the equation: \(2y + x = 430\).
Step 2 ：We also know that the area \(A\) of the rectangle is given by: \(A = xy\). We want to maximize \(A\).
Step 3 ：To do this, we can express \(y\) in terms of \(x\) using the first equation, substitute this into the second equation, and then find the derivative of \(A\) with respect to \(x\). Setting this derivative equal to zero will give us the value of \(x\) that maximizes \(A\).
Step 4 ：We can then substitute this value back into the first equation to find the corresponding value of \(y\).
Step 5 ：From the equation \(2y + x = 430\), we can express \(y\) as \(y = 215 - x/2\).
Step 6 ：Substitute \(y\) into the area equation \(A = xy\), we get \(A = x*(215 - x/2)\).
Step 7 ：Find the derivative of \(A\) with respect to \(x\), we get \(A' = 215 - x\).
Step 8 ：Setting \(A' = 0\), we get \(x = 215\).
Step 9 ：Substitute \(x = 215\) back into the equation \(y = 215 - x/2\), we get \(y = 107.5\).
Step 10 ：The solution to the optimization problem is \(x = 215\) feet and \(y = 107.5\) feet. This means that the length of the side parallel to the store should be 215 feet and the length of the side perpendicular to the store should be 107.5 feet in order to maximize the total area of the outdoor enclosure.
Step 11 ：Final Answer: The length of the sides parallel to the store is \(\boxed{215}\) feet and the length of the sides perpendicular to the store is \(\boxed{107.5}\) feet.
