# SOLUTION: a carpenter is building a rectangular room with a fixed perimeter of 128 ft. what dimensions would yield the maximum area? what is the maximum area?

Algebra ->  Algebra  -> Test -> SOLUTION: a carpenter is building a rectangular room with a fixed perimeter of 128 ft. what dimensions would yield the maximum area? what is the maximum area?      Log On

 Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help! Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!

 Test Calculators and Practice Answers archive Word Problems Lessons In Depth

 Click here to see ALL problems on test Question 402284: a carpenter is building a rectangular room with a fixed perimeter of 128 ft. what dimensions would yield the maximum area? what is the maximum area?Answer by solver91311(16868)   (Show Source): You can put this solution on YOUR website! Let's solve this one in general, that is for any given perimeter. Let P represent the fixed perimeter. Let w represent the width of the rectangle. Let l represent the length of the rectangle. The perimeter of a rectangle is: So The area of a rectangle is the length times the width so a function for the area in terms of the width is: Algebra Solution: The area function is a parabola, opening downward, with vertex at: Since the parabola opens downward, the vertex represents a maximum value of the area function. The value of the width that gives this maximum value is one-fourth of the given perimeter. Therefore, the shape must be a square, and the area is the width squared. Calculus Solution: The area function is continuous and twice differentiable over its domain, therefore there will be a local extrema wherever the first derivative is equal to zero and that extreme point will be a maximum if the second derivative is negative at that point. Therefore the maximum area is obtained when And that maximum area is: John My calculator said it, I believe it, that settles it