Question

Carl 在探索洞穴时发现一批石头： $5$ 磅重的石头每块价值 $ $14$ ， $4$ 磅重的石头每块价值 $ $11$ ， $1$ 磅重的石头每块价值 $ $2$ 。每种大小至少有 $20$ 块。他最多能携带 $18$ 磅。按美元计，他最多能从洞穴中带出价值多少的石头？

While exploring a cave, Carl comes across a collection of $5$ -pound rocks worth $ $14$ each, $4$ -pound rocks worth $ $11$ each, and $1$ -pound rocks worth $ $2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

$48$

$49$

$50$

$51$

$52$

答案：C

知识点：最优化分类讨论

难度评级：1020

解答：

三种石头每磅分别价值 $ $2.80,$ $ $2.75,$ 和 $ $2$ 。对每一种可能的 $5$ 磅石头数量，尽量装入 $4$ 磅石头，再用 $1$ 磅石头填满剩余容量。取 $0,1,2,$ 或 $3$ 块 $5$ 磅石头时，最大价值分别为 $ $48,$ $ $49,$ $ $50,$ 和 $ $48,$

因此，两块 $5$ 磅石头和两块 $4$ 磅石头恰好用满 $18$ 磅，最大价值是 $ $50.$ 所以正确答案是 C。

The rocks are worth $ $2.80,$ $ $2.75,$ and $ $2$ per pound, respectively. For each possible number of $5$ -pound rocks, use as many $4$ -pound rocks as fit and fill any leftover capacity with $1$ -pound rocks. Taking $0,1,2,$ or $3$ of the $5$ -pound rocks gives maximum values $ $48,$ $ $49,$ $ $50,$ and $ $48,$ respectively.

Thus two $5$ -pound and two $4$ -pound rocks use all $18$ pounds, and the maximum value is $ $50.$ Thus, the correct answer is C.

2018 AMC 12A 第 2 题

答案：C

难度评级：1020

解答：

其他年份的第 2 题