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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 \(y\)

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by VJesus12 » Thu Jun 04, 2020 7:35 am

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Answers

A

B

C

D

E

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Difficulty—

\(P, Q\) 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. \(\dfrac{\sqrt{x^2+y^2}}{z}\)

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

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

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

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

[spoiler]OA=C[/spoiler]

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Re: \(P, Q\) and \(R\) are located in a flat region of a certain state. \(Q\) is \(x\) miles due east of \(P\) and \(y\)

by Jay@ManhattanReview » Fri Jun 05, 2020 12:49 am
VJesus12 wrote: ↑
Thu Jun 04, 2020 7:35 am
 \(P, Q\) 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. \(\dfrac{\sqrt{x^2+y^2}}{z}\)

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

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

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

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

[spoiler]OA=C[/spoiler]

Source: GMAT Prep
See the image below.
Picture1.png
The distance from QR and RP = y + √(x^2 + y^2)

Thus, the time taken to cover the distance [y + √(x^2 + y^2)] at the rate of z mph = [y + √(x^2 + y^2)]/z hours

The correct answer: C

Hope this helps!

-Jay
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Re: \(P, Q\) and \(R\) are located in a flat region of a certain state. \(Q\) is \(x\) miles due east of \(P\) and \(y\)

by Scott@TargetTestPrep » Mon Jun 08, 2020 1:02 pm
VJesus12 wrote: ↑
Thu Jun 04, 2020 7:35 am
 \(P, Q\) 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. \(\dfrac{\sqrt{x^2+y^2}}{z}\)

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

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

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

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

[spoiler]OA=C[/spoiler]

Solution:

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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