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：体积相似比与比例

难度评级：1800

解答：

在高度比例为 $t$ 处，水平截面正方形边长为 $1 + 2t$ ，所以装到高度 $t$ 的体积为 $\int_0^t (1 + 2u)^2\,du$ 。当 $\left(t = \tfrac{1}{2}\right)$ 时体积为 $\tfrac{7}{6}$ ，总体积为 $\tfrac{13}{3}$ 。剩余体积为 $\tfrac{13}{3} - \tfrac{7}{6} = \tfrac{19}{6}$ 是前一部分的 $\tfrac{19}{7}$ 倍，因此还需 $35 \cdot \tfrac{19}{7} = 95$ 分钟。

所以正确答案是 D。

At height fraction $t$ the square cross-section has side $1 + 2t,$ so the volume filled up to height $t$ is $\int_0^t (1 + 2u)^2\,du.$ Up to the midline $\left(t = \tfrac{1}{2}\right)$ this is $\tfrac{7}{6},$ and the full volume is $\tfrac{13}{3}.$ The remaining volume is $\tfrac{13}{3} - \tfrac{7}{6} = \tfrac{19}{6},$ which is $\tfrac{19}{7}$ times the first part. So the remainder takes $35 \cdot \tfrac{19}{7} = 95$ more minutes.

Thus, the correct answer is D.

2025 AMC 12B 第 15 题

答案：D

难度评级：1800

解答：

其他年份的第 15 题