Question

两个直圆柱体体积相同。第二个圆柱体的半径比第一个多 $10\%$ 。两个圆柱体的高之间有什么关系？

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?

第二个高比第一个少 $10\%$ 。

The second height is $10\%$ less than the first.

第一个高比第二个多 $10\%$ 。

The first height is $10\%$ more than the second.

第二个高比第一个少 $21\%$ 。

The second height is $21\%$ less than the first.

第一个高比第二个多 $21\%$ 。

The first height is $21\%$ more than the second.

第二个高是第一个的 $80\%$ 。

The second height is $80\%$ of the first.

答案：D

知识点：圆柱体积百分数

难度评级：1220

解答：

设两个圆柱的半径和高分别为 $r_1$ 、 $h_1$ 与 $r_2$ 、 $h_2$ 。

已知且 $r_2 = \dfrac{11}{10}r_1$ $\pi r_1^2h_1 = \pi r_2^2h_2.$

代入得到所以因此第一个圆柱的高比第二个高百分之二十一。 $r_1^2h_1 = \dfrac{121}{100}r_1^2h_2,$ $h_1 = \dfrac{121}{100}h_2.$

所以正确答案是 D。

Let $r_1$ and $h_1$ be the radius and height of the first cylinder and similarly define $r_2$ and $h_2$ for the second cylinder.

We know that $r_2 = \dfrac{11}{10}r_1$ and $\pi r_1^2h_1 = \pi r_2^2h_2.$

Substituting and simplifying gives us $r_1^2h_1 = \dfrac{121}{100}r_1^2h_2,$ which tells us that $h_1 = \dfrac{121}{100}h_2.$

Thus, D is the correct answer.

2015 AMC 10A 第 9 题

答案：D

难度评级：1220

解答：

其他年份的第 9 题