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P, Q and R are located in a flat region of a certain state.

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P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 18 Oct 2008, 21:04
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P, Q and R are located in a flat region of a certain state. Q is x miles due east of P and y miles due north of R. Each pair of points is connected by a straight road. What is the number of hours needed to drive from Q to R and then from R to P at a constant rate of z miles per hour, in terms of x, y and z? (Assume x, y, and z are positive)

A. \(\frac{\sqrt{x^2+y^2}}{z}\)

B. \(\frac{x+\sqrt{x^2+y^2}}{z}\)

C. \(\frac{y+ \sqrt{x^2+y^2}}{z}\)

D. \(\frac{z}{x+\sqrt{x^2+y^2}}\)

E. \(\frac{z}{y+\sqrt{x^2+y^2}}\)
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Jun 2014, 11:34, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 18 Oct 2008, 22:39
Poots wrote:
P, Q and R are located in a flat region of a certain state. Q is x miles due east of P and y miles due north of R. Each pair of points is connected by a straight road. What is the number of hours needed to drive from Q to R and then from R to P at a constant rate of z miles per hour, in terms of x, y and z? (Assume x, y, and z are positive)

a. √x² + y² / z
b. x + √ x² + y² / z
c. y + √x² + y² / z
d. z / x + √x² + y²
e. z / y + √x² + y²


RP= (x^2+y^2)^1/2
QR=y

speed =z

total time =total dist /speed = (x^2+y^2)^1/2 + y / z
IMO C
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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 22 Oct 2008, 17:26
Thanks spriya.

Does anyone have another explanation for this?

Thanks!

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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 22 Oct 2008, 17:45
Thanks LS,

I guess the more detailed follow up question would be: How do you get this for RP?

spriya wrote:
RP= (x^2+y^2)^1/2

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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 22 Oct 2008, 17:52
Oh ok,

If you draw the points, you will get PQR as a right angled traingle with sides as
x, y, sqrt(x^2 + y^2)

And we need to the find the time required to travel the distances y and sqrt(x^2+y^2)

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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 22 Oct 2008, 18:03
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Epiphany! Thanks LS.

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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 21 Jun 2014, 08:56
are the answer choices correctly written here? My answer would be (x^2+y^2)^1/2 + y / z
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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 21 Jun 2014, 11:35
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MensaNumber wrote:
are the answer choices correctly written here? My answer would be (x^2+y^2)^1/2 + y / z


Edited the answer choices.

P, Q and R are located in a flat region of a certain state. Q is x miles due east of P and y miles due north of R. Each pair of points is connected by a straight road. What is the number of hours needed to drive from Q to R and then from R to P at a constant rate of z miles per hour, in terms of x, y and z? (Assume x, y, and z are positive)

A. \(\frac{\sqrt{x^2+y^2}}{z}\)

B. \(\frac{x+\sqrt{x^2+y^2}}{z}\)

C. \(\frac{y+ \sqrt{x^2+y^2}}{z}\)

D. \(\frac{z}{x+\sqrt{x^2+y^2}}\)

E. \(\frac{z}{y+\sqrt{x^2+y^2}}\)

Points are located as shown below:

P----Q
------
------
-----R

So we have the right triangle: \(PQ=x\) and \(QR=y\). \(PR=hypotenuse=\sqrt{x^2+y^2}\).

Total distance to cover: \(y+\sqrt{x^2+y^2}\) --> \(time=\frac{distance}{rate}=\frac{y+\sqrt{x^2+y^2}}{z}\).

Answer: C.
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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 11 Sep 2017, 00:24
It forms a right angle triangle
Base (x) (PQ) and Height (y)(QR)

Hypotenus(PR) sqrt (x^2+y^2)

So total distance: y+ sqrt (x^2+y^2)/z

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Re: P, Q and R are located in a flat region of a certain state. [#permalink]

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New post 11 Sep 2017, 16:00
Poots wrote:
P, Q and R are located in a flat region of a certain state. Q is x miles due east of P and y miles due north of R. Each pair of points is connected by a straight road. What is the number of hours needed to drive from Q to R and then from R to P at a constant rate of z miles per hour, in terms of x, y and z? (Assume x, y, and z are positive)

A. \(\frac{\sqrt{x^2+y^2}}{z}\)

B. \(\frac{x+\sqrt{x^2+y^2}}{z}\)

C. \(\frac{y+ \sqrt{x^2+y^2}}{z}\)

D. \(\frac{z}{x+\sqrt{x^2+y^2}}\)

E. \(\frac{z}{y+\sqrt{x^2+y^2}}\)


We see that P, Q, and R form the vertices of a right triangle, with Q as the vertex of the right angle. Furthermore, PQ = x and QR = y are the legs of the right triangle, and RP is the hypotenuse of the right triangle.

Thus, if we let RP = n, then by the Pythagorean theorem we have:

n^2 = x^2 + y^2

n = √( x^2 + y^2)

Since time = distance/rate, it takes y/z hours to drive from Q to R and √( x^2 + y^2)/z hours to drive from R to P.

So, the total driving time is [y + √( x^2 + y^2)]/z.

Answer: C
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Re: P, Q and R are located in a flat region of a certain state.   [#permalink] 11 Sep 2017, 16:00
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