In the figure above, car A and car B travel around a circular park with an area of 1256 square miles. Both cars leave from the same point, the START location shown in the figure. Car A travels counter-clockwise at 60 miles/hour and car B travels clockwise at 40 miles/hour. Which of the following is closest to the number of minutes it takes for the cars to meet? Use 3.14 for π.

A. 30 B. 50 C. 75 D. 125 E. 180

(C) To visualize this question, think about a circle as just a line segment joined at its two ends. Let’s first cut the circle at the starting point and make it in to a straight line. Notice that cars A and B are at opposite ends of the line traveling toward each other. A [ ______________________ ] B

We know that the area of the circular park is 1256 square miles and that the area of a circle is equal to πr². So, 1256 = πr².

Divide by 3.14, our approximation of π. 1256/3.14 = 400 = r². r = 20 miles.

Now, solve for the circumference of the park: C = 2πr. Plugging 20 into the equation, we get: C[/i] = 2 × 3.14 × 20 = 125.6 miles.

To calculate how long it takes to meet, combine the speeds of car A and car B to see how long it would take them to travel 125.6 miles.

Car A and car B are traveling toward each other at the rate of 40 + 60 = 100 miles per hour (because they are heading toward each other, we add their speeds together). At 100 mph, it will take them 1.256 hours to travel 125.6 miles and meet up. Multiply 1.256 hours by 60 (because there are 60 minutes in one hour, and we are looking for the answer in minutes): 1.256 × 60, which is approximately 75 minutes.

The correct answer is choice (C). -------------

Why is it assumed that both cars complete the full circumference? Isn't it possible that they meet somewhere along the track?

They do meet somewhere along the track. We do NOT assume that EACH car completes the circumference.

Since they must meet somewhere on the circumference, each car travels from the starting point till the meeting point. Clearly, these distances make up the circumference:

(The path of car A is marked in blue and the path of car B is marked in red).

They start from the same point and move towards each other along the circumference. It's inevitable that they meet. When they do - each car will travel a path of an arc from the starting point till the meeting point. These two arcs make the circumference, since the edges are the same points.

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