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Problem # 1 : https://prikachi.com/images.php?images/252/7476252Z.jpg
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2 problem solving difficulties
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The sides of a 30-60-90 triangle are proportioned x : x√3 : 2x.
Whenever you see √3 in a geometry question, look for -- or DRAW -- a 30-60-90 triangle.
Given (-√3, 1), y=1 and |x|=√3 represent the legs of a 30-60-90 triangle in which the sides are 1:√3:2.
Thus, we get the following drawing:
The correct answer is B.
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We can PLUG IN THE ANSWERS, which represent the number of hours that the boat took to travel downstream.A boat traveled upstream a distance of 90 miles at an average speed of
(v-3) miles per hour and then traveled the same distance downstream at an
average speed of (v+3) miles per hour. If the trip upstream took half an
hour longer than the trip downstream, how many hours did it take the boat
to travel downstream?
a) 2.5
b) 2.4
c) 2.3
d) 2.2
e) 2.1
When the correct answer choice is plugged in, the time upstream will be 1/2 hour longer than the time downstream.
The most likely answer choice is A -- the only option that divides evenly into 90.
Answer choice A: 2.5 hours to travel downstream.
Rate downstream = d/t = 90/(2.5) = 36 miles per hour.
Thus, v+3 = 36, implying that v=33.
Rate upstream = v-3 = 33-3 = 30 miles per hour.
Time upstream = d/r = 90/30 = 3 hours.
Time upstream - time downstream = 3-2.5 = .5.
Success!
The correct answer is A.
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I like to begin with a "word equation." We can write:A boat travelled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then travelled downstream at an average speed of (V+3) miles per hour. If the trip upstream took half an hour longer than the trip downstream, how many hours did it take the boat to travel downstream?
A) 2.5
B) 2.4
C) 2.3
D) 2.2
E) 2.1
travel time upstream = travel time downstream + 1/2
Time = distance/rate
So, we can replace elements in our word equation to get:
90/(v-3) = 90/(v+3) + 1/2
Now solve for v (lots of work here)
.
.
.
v = 33
So, travel time downstream = 90/(v+3)
= 90/(33+3)
= 90/36
= 5/2
= 2 1/2 hours
Cheers,
Brent
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- GMATinsight
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Another logical and very short method to solve this question is as follows
Slope of line OP = -(1/√3)
Product of slopes of perpendicular lines = -1
therefore slope of line OQ = √3/1
Since OP and OQ are both radii and the co ordinates of P are representing the slope without canceling any common factor between numerator and denominator therefore the slope of OQ Also will also represent the co-ordinates of the point Q [Furthermore this can also be verified by taking the distance between O and Q which becomes equal to radius=2 of the circle]
Slope of OQ = t/s = √3/1
therefore (s,t) = (1,√3)
therefore,
s = 1
Answer Option B
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