Question

一个容器底部是 $1 \times 1$ 的正方形，顶部开口是 $3 \times 3$ 的正方形，四个侧面是全等的梯形，如图所示。从空容器开始，一根以恒定速率出水的水管用 $35$ 分钟把容器装到梯形侧面的中线高度。

还需要多少分钟才能把容器剩余部分装满？

A container has a $1 \times 1$ square bottom, a $3 \times 3$ open square top, and four congruent trapezoidal sides, as shown. Starting when the container is empty, a hose that runs water at a constant rate takes $35$ minutes to fill the container up to the midline of the trapezoids.

How many more minutes will it take to fill the remainder of the container?

$70$

$85$

$90$

$95$

$105$

答案：D

知识点：长度、面积与体积的缩放关系体积相似

难度评级：1660

解答：

这个容器是一个正方台：高度比例为 $t$ 的水平截面边长为 $1 + 2t$ 。把侧面向上延长到顶点，则到边长为 $w$ 的截面为止的体积按 $w^3$ 缩放。因此整个容器有 $3^3 - 1^3 = 26$ 份，到中线（边长 $2$ ）的部分有 $2^3 - 1^3 = 7$ 份，剩余部分有 $3^3 - 2^3 = 19$ 份。 $7$ 份用时 $35$ 分钟，所以每份 $5$ 分钟。剩下 $19$ 份需要 $95$ 分钟。因此正确答案是 D。

The container is a square frustum: a horizontal slice at height fraction $t$ has side $1 + 2t.$ Extend the sides up to their apex, and the volume out to where the side length is $w$ scales as $w^3.$ So the whole container is $3^3 - 1^3 = 26$ parts, the piece up to the midline (side $2$ ) is $2^3 - 1^3 = 7$ parts, and the rest is $3^3 - 2^3 = 19$ parts. Those $7$ parts take $35$ minutes, so each part is $5$ minutes. The remaining $19$ parts take $95$ minutes. Thus, D is the correct answer.

2025 AMC 10B 第 19 题

答案：D

难度评级：1660

解答：

其他年份的第 19 题