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24 Jun 2022, 01:30
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Pamela needs to drive from Town A to Town B. She can take Path 1, which is a straight road connecting the two towns, and drive at a speed of 100 kilometers per hour. Alternatively, she can take Path 2, in which she first drives on a straight road from Town A to Town C which is 100√3 kilometers away and then takes the perpendicular road that connects Town C to Town B. If she can drive at a speed of 100√3 kilometers per hour on Path 2 and the road that connects Town A to Town C is at an angle of 30∘ to the road that connects Town A to Town B, by taking which path will she reach Town B earlier and approximately how much time will she save by taking that path instead of the other path?
A) Path 1, 26 minutes
B) Path 1, 34 minutes
C) Path 1, 1 hour 34 minutes
D) Path 2, 26 minutes
E) Path 2, 1 hour
Are You Up For the Challenge: 700 Level Questions
A) Path 1, 26 minutes
B) Path 1, 34 minutes
C) Path 1, 1 hour 34 minutes
D) Path 2, 26 minutes
E) Path 2, 1 hour
Are You Up For the Challenge: 700 Level Questions
24 Jun 2022, 09:01
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Pamela needs to drive from Town A to Town B. She can take Path 1, which is a straight road connecting the two towns, and drive at a speed of 100 kilometers per hour. Alternatively, she can take Path 2, in which she first drives on a straight road from Town A to Town C which is 100√3 kilometers away and then takes the perpendicular road that connects Town C to Town B. If she can drive at a speed of 100√3 kilometers per hour on Path 2 and the road that connects Town A to Town C is at an angle of 30∘ to the road that connects Town A to Town B, by taking which path will she reach Town B earlier and approximately how much time will she save by taking that path instead of the other path?
A) Path 1, 26 minutes
B) Path 1, 34 minutes
C) Path 1, 1 hour 34 minutes
D) Path 2, 26 minutes
E) Path 2, 1 hour
A) Path 1, 26 minutes
B) Path 1, 34 minutes
C) Path 1, 1 hour 34 minutes
D) Path 2, 26 minutes
E) Path 2, 1 hour
We are told that angle C is 90 degrees and that angle A is 30 degrees. We are looking at a 30-60-90 triangle.
We are told that AC = \(100\sqrt{3}\), so BC = 100 and AB = 200.
Path 1 is 200km long. At a rate of 100kph, that would take 2 hours.
Path 2 is \(100\sqrt{3}+100\)km = roughly 170+100 = 270km long. At a rate of 170kph, that will take a little more than 1.5 hours.
Path 2 is faster by a little less than half an hour.
Answer choice D.
Like most geometry questions (and of all difficulty levels), we can get to the right answer by ballparking things like \(\sqrt{3}\) and skipping all the annoying conversions.
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