1957 AMC 12 第 6 题

Question

用一块

10

英寸乘

14

英寸的长方形金属片制作一个无盖盒子：从每个角剪去一个边长为

x

英寸的正方形，再把留下的凸出部分向上折起并焊接接缝。所得盒子的体积为：

An open box is constructed by starting with a rectangular sheet of metal

10

in. by

14

in. and cutting a square of side

x

inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:

140x-48x^2+4x^3

140x+48x^2+4x^3

140x+24x^2+x^3

140x-24x^2+x^3

以上都不是

none of these

折起后，盒高为

x

，底面的每个边长都减少

2x

After folding, the height is

x

and each base dimension loses

2x

将体积写成

x(10-2x)(14-2x)

Write the volume as

x(10-2x)(14-2x)

盒高为

x

，底面的两个边长分别为

10-2x

和

14-2x

。因此

\begin{aligned} V&=x(10-2x)(14-2x)\\ &=140x-48x^2+4x^3 \end{aligned}\text{。}

因此，正确答案是 A。

The box has height

x

and base dimensions

10-2x

and

14-2x.

Hence

\begin{aligned} V&=x(10-2x)(14-2x)\\ &=140x-48x^2+4x^3. \end{aligned}

Thus, the correct answer is A.

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