Pure Maths Question

Question

Please help me to solve the following question:

Show that a rectangle with a given perimeter achieves maximum area only if it is a square.

Answer 1

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假設矩形的周長是 A, 令此矩形的其中一邊長是X,則另一邊長必為(A/2-X)

矩形的面積則為:X*(A/2-X)=X*A/2-X^2

利用"配方法"則上式可寫成-(X^2-AX/2+(A/4)^2)+0.5A^2=-(X-A/4)^2+A^2/16

當X=A/4的時候,可以得到最大面積=A^2/16

此時矩形的形狀即為正方形(因為邊長都是A/4)

Answer 2

你們可以用Derivative Method解答嗎？

Answer 3

Terry +1! Nice job! you didn't use "derivative" but "completing the square" (algebraic method).

2012-11-01 07:47:19 補充：

calculus method:

assume x and y are the two side of the rectangular, hence

2x + 2y = A (A is fixed perimeter)

==> x = 1/2(A- 2y)

2012-11-01 07:50:30 補充：

area of rectangular = xy = 1/2(A-2y)y = Ay/2 -y^2

for area to be the maximum, d(area)/dy = 0

hence, A/2 − 2y = 0 ==> y = A/4, so x=y=A/4, a square !

2012-11-03 08:55:51 補充：

Using what method to solve a problem depends upon the level of schooling. If a person can understand the meaning of derivatives, then why not uses a more advance way to solve a problem.

2012-11-03 08:57:34 補充：

However, I feel, no matter what level of schooling you are in, a person shall always know a lower level method. After all, it is a foundation for more advanced method.