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考题7-8 2015 AMC 12A

2018-08-22 重点归纳

AMC12是针对高中学生的数学测验，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。其主要目的在于激发学生对数学的兴趣，参予AMC12的学生应该不难发现测验的问题都很具挑战性，但测验的题型都不会超过学生的学习范围。这项测验希望每个考生能从竞赛中享受数学。那么接下来跟随小编来看一下AMC12的官方真题以及官方解答吧：

Problem 7

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?

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Solution

Let the radius of the first cylinder be $r_1$ and the radius of the second cylinder be $r_2$ . Also, let the height of the first cylinder be $h_1$ and the height of the second cylinder be $h_2$ . We are told $\pi r_1^2h_1=\pi r_2^2h_2$ Substituting the first equation into the second and dividing both sides by $\pi$ , we get $r_1^2h_1=\frac{121r_1^2}{100}h_2\implies h_1=\frac{121h_2}{100}.$ Therefore, $\boxed{\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}}$

Problem 8

The ratio of the length to the width of a rectangle is $4$ : $3$ . If the rectangle has diagonal of length $d$ , then the area may be expressed as $kd^2$ for some constant $k$ . What is $k$ ?

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Solution

Let the rectangle have length $4x$ and width $3x$ . Then by $3-4-5$ triangles (or the Pythagorean Theorem), we have $d = 5x$ , and so $x = \dfrac{d}{5}$ . Hence, the area of the rectangle is $amc数学竞赛官网$ , so the answer is $amc数学竞赛官网$