Anonymous
Anonymous asked in 社會與文化語言 · 7 years ago

Pure Maths 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.

3 Answers

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  • 7 years ago
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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)

  • 7 years ago

    你們可以用Derivative Method解答嗎?

  • 7 years ago

    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.

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