106 學年度台綜大微積分 A 第 10 題

題目

10. (10 points) A rectangular box is formed by cutting four equal corners from a square of side

3

and then folding up (see the figure below). Find the maximum possible volume of the box.

設每個角切去邊長 $x$ 的正方形，則

V=x(3-2x)^2,\qquad 0<x<\frac32.

由 $V'(x)=(3-2x)(3-6x)$ ，最大值在 $x=\frac12$ 取得，故

\boxed{V_{\max}=2}.