Question

一个等边三角形最初被涂成黑色。每次改变时，每个黑色三角形的中间四分之一变成白色。经过五次改变后，仍为黑色的部分占原三角形面积的几分之几？

An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?

$\dfrac{1}{1024}$

$\dfrac{15}{64}$

$\dfrac{243}{1024}$

$\dfrac14$

$\dfrac{81}{256}$

答案：C

知识点：等比数列面积比

难度评级：1140

小提示：

改变一次后，中间四分之一变白，所以黑色面积剩下 $\dfrac34$

After one change the middle fourth turns white, so $\dfrac34$ of the black area remains

大提示：

每次改变都会把黑色面积乘以 $\dfrac34$ ，所以五次后乘以 $\left(\dfrac34\right)^5$

Each change multiplies the black area by $\dfrac34,$ so after five changes multiply by $\left(\dfrac34\right)^5$

解答：

每一次改变都会让当前黑色面积的 $\dfrac34$ 保持黑色。五次改变后，黑色面积比例为 $\left(\dfrac34\right)^5 = \dfrac{243}{1024}\text{。}$

所以正确答案是 C。

Each change leaves $\dfrac34$ of the current black area black. After five changes the black fraction is $\left(\dfrac34\right)^5 = \dfrac{243}{1024}.$

Thus, the correct answer is C .

1991 AMC 8 第 25 题

答案：C

难度评级：1140

小提示：

大提示：

解答：

其他年份的第 25 题