AMC12 2025 B

AMC12 2025 B · Q15

AMC12 2025 B · Q15. It mainly tests Arithmetic misc, 3D geometry (volume).

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?

一个容器底部是$1\times1$正方形，顶部是$3\times3$开口正方形，有四个全等的梯形侧面，如图所示。从容器为空开始，一根以恒定速率注水的软管需要35分钟将容器填充到梯形中线高度。填充容器剩余部分还需要多少分钟？

(A) 70 70

(B) 85 85

(D) 95 95

(E) 105 105

Answer

Correct choice: (D)

正确答案：（D）

Solution

Extend the edges pointing downwards to converge at a point $A$ to form a square pyramid. Consider 3 square pyramids, the large one formed by the top vertices of the original figure and $A$, the middle one formed by the medians running through the sides of the original figure and point $A$, and the smaller one formed by the bottom vertices of the original figure and point $A$. Note that all pyramids are similar since they are all essentially scaled by a certain factor. The median length is $\frac{3+1}{2}=2$ Using side length to volume ratios, find that the volumes must have ratios $1:8:27$ Then, you get that the ratio of the volume thus filled to the volume that we must fill is equivalent to $8-1:27-8 = 7:19$. Thus, it will take $\frac{19}{7}$ more time to fill the remaining volume giving us an answer of $\frac{19}{7} * 35 = \boxed{\textbf{(D) }95}$

将向下延伸的边缘延长到点$A$相交，形成一个正方锥。考虑3个正方锥：由原图顶部顶点和$A$形成的大锥，由原图侧面中线和点$A$形成的中锥，由原图底部顶点和点$A$形成的小锥。注意所有锥体都是相似的，因为它们本质上按一定因子缩放。中线长度是$\frac{3+1}{2}=2$。使用边长到体积的比例，发现体积比例为$1:8:27$。然后，已填充体积与剩余体积的比例等价于$8-1:27-8=7:19$。因此，填充剩余体积需要$\frac{19}{7}$倍时间，给出答案$\frac{19}{7}\times35=\boxed{\textbf{(D) }95}$

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