A farmer has 180180 feet of fencing to construct a rectangular pen up against the straight side of a barn, using the barn for one side of the pen. The length of the barn is 140140 feet. Determine the dimensions of the rectangle of maximum area that can be enclosed under these conditions. (Hint: Be mindful of the domain of the function you are maximizing.)
Question
A farmer has 180180 feet of fencing to construct a rectangular pen up against the straight side of a barn, using the barn for one side of the pen. The length of the barn is 140140 feet. Determine the dimensions of the rectangle of maximum area that can be enclosed under these conditions. (Hint: Be mindful of the domain of the function you are maximizing.)
Solution
Sure, let's solve this step by step:
Step 1: Define the variables Let's denote the length of the rectangle (parallel to the barn) as x and the width of the rectangle (perpendicular to the barn) as y.
Step 2: Set up the equations We know that the total length of the fencing is 180 feet. Since the barn is used as one side of the rectangle, the total length of the fencing is equal to 2y + x. So, we have the equation: 2y + x = 180
Step 3: Express one variable in terms of the other From the equation above, we can express x in terms of y: x = 180 - 2y
Step 4: Set up the area function The area A of the rectangle is given by the product of its length and width, so: A = xy Substitute x from step 3 into the area function: A = y(180 - 2y) = 180y - 2y^2
Step 5: Find the maximum area The maximum area is found by taking the derivative of the area function with respect to y and setting it equal to zero: A' = 180 - 4y = 0 Solving for y gives y = 45 feet.
Step 6: Find the corresponding x value Substitute y = 45 into the equation from step 3 to find x: x = 180 - 2*45 = 90 feet.
So, the dimensions of the rectangle of maximum area that can be enclosed under these conditions are 90 feet by 45 feet.
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