AMC10 2009 B

AMC10 2009 B · Q14

AMC10 2009 B · Q14. It mainly tests Exponents & radicals, Interest / growth (simple).

On Monday, Millie puts a quart of seeds, 25% of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only 25% of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?

周一，Millie向鸟食器中放入一夸脱种子，其中25%是小米。此后每天她再添加一夸脱相同混合的种子，而不移除剩余的种子。每天鸟儿只吃掉食器中25%的小米，但吃掉所有其他种子。在Millie放入种子后哪一天，鸟儿会发现食器中超过一半的种子是小米？

(A) Tuesday 星期二

(B) Wednesday 星期三

(D) Friday 星期五

(E) Saturday 星期六

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): On Monday, day 1, the birds find $\frac{1}{4}$ quart of millet in the feeder. On Tuesday they find \[ \frac{1}{4}+\frac{3}{4}\cdot\frac{1}{4} \] quarts of millet. On Wednesday, day 3, they find \[ \frac{1}{4}+\frac{3}{4}\cdot\frac{1}{4}+\left(\frac{3}{4}\right)^2\cdot\frac{1}{4} \] quarts of millet. The number of quarts of millet they find on day $n$ is \[ \frac{1}{4}+\frac{3}{4}\cdot\frac{1}{4}+\left(\frac{3}{4}\right)^2\cdot\frac{1}{4}+\cdots+\left(\frac{3}{4}\right)^{n-1}\cdot\frac{1}{4} =\frac{\left(\frac{1}{4}\right)\left(1-\left(\frac{3}{4}\right)^n\right)}{1-\frac{3}{4}} =1-\left(\frac{3}{4}\right)^n. \] The birds always find $\frac{3}{4}$ quart of other seeds, so more than half the seeds are millet if $1-\left(\frac{3}{4}\right)^n>\frac{3}{4}$, that is, when $\left(\frac{3}{4}\right)^n<\frac{1}{4}$. Because $\left(\frac{3}{4}\right)^4=\frac{81}{256}>\frac{1}{4}$ and $\left(\frac{3}{4}\right)^5=\frac{243}{1024}<\frac{1}{4}$, this will first occur on day 5 which is Friday.

答案（D）：在星期一（第 1 天），鸟在喂食器里找到 $\frac{1}{4}$ 夸脱的小米。星期二它们找到 \[ \frac{1}{4}+\frac{3}{4}\cdot\frac{1}{4} \] 夸脱的小米。星期三（第 3 天），它们找到 \[ \frac{1}{4}+\frac{3}{4}\cdot\frac{1}{4}+\left(\frac{3}{4}\right)^2\cdot\frac{1}{4} \] 夸脱的小米。第 $n$ 天它们找到的小米数量（单位：夸脱）为 \[ \frac{1}{4}+\frac{3}{4}\cdot\frac{1}{4}+\left(\frac{3}{4}\right)^2\cdot\frac{1}{4}+\cdots+\left(\frac{3}{4}\right)^{n-1}\cdot\frac{1}{4} =\frac{\left(\frac{1}{4}\right)\left(1-\left(\frac{3}{4}\right)^n\right)}{1-\frac{3}{4}} =1-\left(\frac{3}{4}\right)^n. \] 鸟总能找到 $\frac{3}{4}$ 夸脱的其他种子，所以当 $1-\left(\frac{3}{4}\right)^n>\frac{3}{4}$（即 $\left(\frac{3}{4}\right)^n<\frac{1}{4}$）时，小米会超过种子总量的一半。因为 $\left(\frac{3}{4}\right)^4=\frac{81}{256}>\frac{1}{4}$ 且 $\left(\frac{3}{4}\right)^5=\frac{243}{1024}<\frac{1}{4}$，这种情况首次发生在第 5 天，也就是星期五。

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