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

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

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

OA C

(Note: Choice C should be [y + √( x^2 + y^2)]/z. That is, “y +” should be part of the numerator.)

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