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國三數學的二是函數應用問題

Mar 16th 2014, 12:21

設把繩成 4x 公分和 (88 - 4x) 公分兩段。
設面積和為 A 平方公分。

A = (4x/4)² + [(88 - 4x)/4]²
A = x² + (22 - x)²
A = x² + 484 - 44x + x²
A = 2x² - 44x + 484
A = 2(x² - 22x + 11²) - 2(11)² + 242
A = 2(x - 11)² + 242

因為 2(x - 11)² ≥ 0
所以 A = 2(x - 11)² + 242 ≥ 242

當 2(x - 11)² = 0 時，A 的最小值 = 242
x - 11 = 0
x = 11
4x = 44
88 - 4x = 44

要讓正方形面積為最小時，把繩剪成兩段 44 公分。
此時面積和 = 242 平方公分

參考資料： andrew

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