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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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Updated on: 21 Jun 2014, 11:34
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15% (low)

Question Stats:

84% (02:11) correct 16% (02:33) wrong based on 439 sessions

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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. $$\frac{z}{y+\sqrt{x^2+y^2}}$$

Originally posted by Poots on 18 Oct 2008, 21:04.
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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21 Jun 2014, 11:35
4
2
MensaNumber wrote:
are the answer choices correctly written here? My answer would be (x^2+y^2)^1/2 + y / z

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: $$y+\sqrt{x^2+y^2}$$ --> $$time=\frac{distance}{rate}=\frac{y+\sqrt{x^2+y^2}}{z}$$.

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

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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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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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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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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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22 Oct 2008, 18:03
1
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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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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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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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.

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

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01 Apr 2018, 07:22
Bunuel Please provide a detailed pictorial explanation. Hard to understand how there is a right triangle. 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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01 Apr 2018, 07:51
1
Bunuel Please provide a detailed pictorial explanation. Hard to understand how there is a right triangle. Thanks.

Since the point Q is
1. y miles due north of R
2. x miles due east of P
and are connected via straight roads, the points PQR form a right-angled triangle

Attachment:

Diagram.png [ 3.17 KiB | Viewed 1259 times ]

Hope this helps you!
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P, Q and R are located in a flat region of a certain state.  [#permalink]

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03 Jul 2018, 05:11
Bunuel wrote:
MensaNumber wrote:
are the answer choices correctly written here? My answer would be (x^2+y^2)^1/2 + y / z

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: $$y+\sqrt{x^2+y^2}$$ --> $$time=\frac{distance}{rate}=\frac{y+\sqrt{x^2+y^2}}{z}$$.

Bunuel we are asked to find Q to R and R to P , in other words TWO LINES , why do you use $$PQ=x$$

and where from do you get Y which is not under radical sign ? $$y+\sqrt{x^2+y^2}$$

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

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03 Jul 2018, 08:34
1
dave13 wrote:
Bunuel wrote:
MensaNumber wrote:
are the answer choices correctly written here? My answer would be (x^2+y^2)^1/2 + y / z

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: $$y+\sqrt{x^2+y^2}$$ --> $$time=\frac{distance}{rate}=\frac{y+\sqrt{x^2+y^2}}{z}$$.

Bunuel we are asked to find Q to R and R to P , in other words TWO LINES , why do you use $$PQ=x$$

and where from do you get Y which is not under radical sign ? $$y+\sqrt{x^2+y^2}$$

thank you

You don't read questions carefully enough.

The question asks to find the number of hours need to drive from Q to R and then from R to P at a constant rate of z miles per hour, NOT the distance.

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

Attachment:

Untitled.png [ 4.79 KiB | Viewed 668 times ]

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Re: P, Q and R are located in a flat region of a certain state. &nbs [#permalink] 03 Jul 2018, 08:34
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