AMC12 2014 A

AMC12 2014 A · Q11

AMC12 2014 A · Q11. It mainly tests Linear equations, Rates (speed).

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?

David 从家开车去机场赶飞机。第一小时他开了 $35$ 英里，但意识到如果继续以这个速度，他将迟到 $1$ 小时。他将速度增加 $15$ 英里每小时，继续前往机场，并提前 $30$ 分钟到达。机场离他家多少英里？

(A) 140 140

(B) 175 175

(D) 245 245

(E) 280\qquad 280\qquad

Answer

Correct choice: (C)

正确答案：（C）

Solution

Note that he drives at $50$ miles per hour after the first hour and continues doing so until he arrives. Let $d$ be the distance still needed to travel after $1$ hour. We have that $\dfrac{d}{50}+1.5=\dfrac{d}{35}$, where the $1.5$ comes from $1$ hour late decreased to $0.5$ hours early. Simplifying gives $7d+525=10d$, or $d=175$. Now, we must add an extra $35$ miles traveled in the first hour, giving a total of $\boxed{\textbf{(C) } 210}$ miles.

注意，他在第一小时后以 $50$ 英里每小时开车，并继续这样做直到到达。设 $d$ 为第一小时后仍需行驶的距离。我们有 $\dfrac{d}{50}+1.5=\dfrac{d}{35}$，其中 $1.5$ 来自迟到 $1$ 小时减少到提前 $0.5$ 小时。化简得到 $7d+525=10d$，即 $d=175$。现在，加上第一小时行驶的额外 $35$ 英里，总共 $\boxed{\textbf{(C) } 210}$ 英里。

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