PSAT Math Multiple-Choice Question 619: Answer and Explanation

Question: 619

At 1:00 P.M., a blimp and a hot-air balloon are above the cities of Springfield and Washington, respectively. The two cities are 300 miles apart horizontally. The blimp is moving from Springfield to Washington at a horizontal speed of 10 miles per hour; the balloon is moving from Washington to Springfield at a horizontal speed of 200 miles per hour. The blimp starts at an altitude of 5,000 feet and is descending at a rate of 5 feet per minute; the balloon starts at an altitude of 500 feet and is ascending at a rate of 4 feet per minute. At what time will the blimp and balloon be at the same altitude, to the nearest minute?

• A. 6:20 P.M.
• B. 7:20 P.M.
• C. 8:20 P.M.
• D. 9:20 P.M.

Correct Answer: D

Explanation:

(D) Don't get confused by all of the unnecessary information here! We want to know when the hot-air balloon and the blimp will be at the same altitude. Since altitude deals with only vertical movement, we only care about their vertical movements. The balloon and blimp will be at the same altitude when their vertical positions are equal.

Start with some notation. Let's say that traveling up is in the positive direction. So the balloon is traveling in the positive direction. Let's also say that traveling down is in the negative direction. So the blimp is traveling in the negative direction.

Remember that distance = rate × time.

The blimp's position can be defined as s = 5,000 - 5t.

The balloon's position can be described as s = 500 + 4t.

The blimp and balloon will be at the same altitude when their positions are equal. So we can set the two expressions equal to one another:

5,000 - 5t = 500 + 4t

Add 5t to both sides while subtracting 500 from both sides:

4,500 = 9t

Dividing by 9 tells us that t = 500.

Because our rates were in feet/minute, our time is in minutes. Let's divide by 60 to convert this to hours:

So it takes 8 hours and of the 9th hour. One-third of an hour is 20 minutes since . So it takes 8 hours and 20 minutes. Since the balloon and blimp started moving toward one another at 1:00 P.M., they'll meet at 9:20 P.M., which is choice (D).