Problem 46 - Introductory Algebra for College Students

Question

Two skiers begin skiing along a trail at the same time. The faster skier averages 9 miles per hour and the slower skier averages 6 miles per hour. The faster skier completes the trail \(\frac{1}{4}\) hour before the slower skier. How long is the trail?

Step-by-Step Solution

Answer

The trail is 27 miles long.

Step 1: Notations and Given Information

Let \(T\) be the time it takes for the slower skier to complete the trail in hours, and \(D\) be the distance of the trail in miles. The faster skier completes the trail in \(T-\frac{1}{4}\) hours.

Step 2: Form Equations based on Given Speeds and Times

The speed is distance divided by time, so the distance covered by both skiers can be represented as: For the faster skier, \(D= 9(T-\frac{1}{4})\); For the slower skier, \(D=6T\)

Step 3: Solve the Equations

Setting the expressions for \(D\) equal to each other gives: \(9(T-\frac{1}{4})=6T\). Solving this equation gives \(T=\frac{9}{2}\) hours. Substituting \(T\) into \(D=6T\) gives \(D = 27\) miles.