AMC10 2015 A

AMC10 2015 A · Q9

AMC10 2015 A · Q9. It mainly tests 3D geometry (volume).

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?

两个正圆柱的体积相同。第二个圆柱的半径比第一个圆柱的半径大 $10\%$。问这两个圆柱的高之间有什么关系？

(A) Second height 10% less than first. 第二高度比第一少10%。

(B) First height 10% more than second. 第一高度比第二多10%。

(D) First height 21% more than second. 第一高度比第二多21%。

(E) Second height 80% of first. 第二高度是第一的80%。

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Let $r, h, R, H$ be the radii and heights of the first and second cylinders, respectively. The volumes are equal, so $\pi r^2 h = \pi R^2 H$. Also $R = r + 0.1r = 1.1r$. Thus $\pi r^2 h = \pi (1.1r)^2 H = \pi (1.21r^2)H$. Dividing by $\pi r^2$ yields $h = 1.21H = H + 0.21H$. Thus the first height is 21% more than the second height.

答案（D）：设 $r, h, R, H$ 分别为第一个和第二个圆柱的半径与高。体积相等，因此 $\pi r^2 h = \pi R^2 H$。又有 $R = r + 0.1r = 1.1r$。因此 $\pi r^2 h = \pi (1.1r)^2 H = \pi (1.21r^2)H$。两边同除以 $\pi r^2$ 得 $h = 1.21H = H + 0.21H$。因此第一个圆柱的高度比第二个高出 21%。

Topics

GE.013 3D geometry (volume)