The Biggest Area
Part of the Quadratics & Complex Numbers section of Coddy's Algebra 2 journey. Lesson 44 of 51.
You have 40 metres of fence to go all the way round a rectangle. If the width is w, the two widths and two lengths add up to 40, so one width and one length add up to 20. The length is 20 - w, and the area is A(w) = w(20 - w).
Which width gives the most room? Multiplied out, A(w) = -w^2 + 20w: a parabola opening down, so its vertex is the greatest area. Its zeros are w = 0 and w = 20, where the rectangle is flat and has no area. The vertex is halfway between them, at w = 10, and A(10) = 10(10), which is 100 square metres.
The best rectangle is a 10 by 10 square. That is no accident: for a fixed length of fence, a square always encloses the most area of all rectangles.
Challenge
A farmer has 60 metres of fence for a rectangular pen. A pen w metres wide is 30 - w metres long, so its area is A(w) = w(30 - w) square metres. Find the biggest area A. On the blank board, first write the best width as a w = ... line, then the area as an A = ... line.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Quadratics & Complex Numbers
2Completing the Square
Perfect Square TrinomialsThe Missing NumberCompleting the SquareIrrational AnswersWhen a Is Not 1Recap: Completing the Square5Complex Arithmetic
Adding and SubtractingMultiplyingThe ConjugateDividingA Pure Imaginary BottomRecap: Complex Arithmetic8Lines Meet Parabolas
Two CrossingsJust TouchingNo MeetingA Circle and a LineRecap: Lines and Parabolas