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A bicyclist travels 8 miles due west at a constant speed. Next, she

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A bicyclist travels 8 miles due west at a constant speed. Next, she  [#permalink]

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New post 20 Jul 2017, 21:36
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Question Stats:

21% (03:46) correct 79% (03:09) wrong based on 32 sessions

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A bicyclist travels 8 miles due west at a constant speed. Next, she rides x miles in a straight line in a direction somewhere between north and east, traveling at half the speed. She stops when she is due north of her starting point, at which time she is y miles from her original location. She then rides, at 1/3 of her original speed, due south for x/2 miles, at which point she ends her trip, more than x/3 miles from her starting point. If x and y are integers, how many total miles did she cycle?

(A) 15
(B) 24
(C) 32.5
(D) 33.5
(E) 40

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A bicyclist travels 8 miles due west at a constant speed. Next, she  [#permalink]

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New post 21 Jul 2017, 01:36
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Since the bicyclist, travels 8 miles due west and then moves towards a point northwards of the point from which he started.
He has given the distance traveled in order to reach the point north as x(which is y miles from original location)

So, it basically forms a right angled triangle with a hypotenuse x, sides y, and 8
Applying Pythagoras theorem, \(x^2 = y^2 + 8^2 => x^2 - y^2 = 64\)
which is possible in 2 cases {(x=10,y=6) or (x=17,y=15)} as \((x-y)(x+y) = 64\)

Case 1: The other sides are x=10 and y=6 because 16*4 = 64.
The sum of the distance walked is \(8+x+\frac{x}{2} = 8+6+2 = 16\), which is not available in the answer options.

Case 2: The only other possibility is 17 and 15, where x=17 and y=15.
So, total distance traveled will be \(8+x+\frac{x}{2} = 8+17+\frac{17}{2} = 25+8.5 = 33.5\)(Option D)
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Re: A bicyclist travels 8 miles due west at a constant speed. Next, she  [#permalink]

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New post 21 Jul 2017, 04:56
since x^2 = y^2 + 8^2 we will get x^2-y^2 = 64
or we can write this as (x+y) (x-y) = 64
now this triangle cant be of 30:60:90 because this gives the sides as 3:4:5 and y cant be 6
why? if y is 6 then it means x is 10,...question says that bicyclist travel x/2 and not the whole distance of y thus x/2 will be equal to 5. The rest of the distance must be greater than x/3 or greater than 3...this makes least value of y as 5+3 = 8 and this it isnt possible that y equals to 6.
Only value that suits is 17 nd 15
(17+15) (17-15) =64--->this means x is 17, y 15 and the base 8
Now total distance travel x + x/2 + 8 = 17+8.5+8 = 33.5...D is the answer
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Re: A bicyclist travels 8 miles due west at a constant speed. Next, she   [#permalink] 21 Jul 2017, 04:56
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