AMC12 2015 A
AMC12 2015 A · Q7
AMC12 2015 A · Q7. 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?
两个直圆柱体体积相同。第二圆柱体的半径比第一圆柱体的半径多10%。这两个圆柱体的高度关系是怎样的？
(A)
The second height is 10% less than the first.
第二个高度比第一个少 10%。
(B)
The first height is 10% more than the second.
第一个高度比第二个多 10%。
(C)
The second height is 21% less than the first.
第二个高度比第一个少 21%。
(D)
The first height is 21% more than the second.
第一个高度比第二个多 21%。
(E)
The second height is 80% of the first.
第二个高度是第一个的 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 $\pi r^2 h=\pi(1.1r)^2H=\pi(1.21r^2)H$. 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$R=r+0.1r=1.1r$。于是$\pi r^2 h=\pi(1.1r)^2H=\pi(1.21r^2)H$。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21%。
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