Justin must travel from Summersville to Clayton. He can drive 15 mi through the mountains at 45 mph . Or hecan drive east and then north on superhighways at 60 mph . The alternative route forms a right angle as shown in the diagram. The eastern leg is 3 mi less than the northern leg. (a) Find the total distance Justin would travel in going the alternative route. (b) If Justin wants to minimize the time of the trip, which route should he take? Clayton

1. Define the variables for the superhighway route. Let the northern leg be \( x \) miles. Then the eastern leg is \( x-3 \) miles. 2. Since the direct mountain route from Summersville to Clayton is 15 miles, and the superhighway route forms a right triangle with these legs, the Pythagorean theorem gives \[ (x-3)^2 + x^2 = 15^2. \] 3. Expand and simplify the equation: \[ (x-3)^2 + x^2 = x^2 - 6x + 9 + x^2 = 2x^2 - 6x + 9. \] Hence, \[ 2x^2 - 6x + 9 = 225. \] 4. Subtract 225 from both sides: \[ 2x^2 - 6x + 9 - 225 = 0 \quad \Longrightarrow \quad 2x^2 - 6x - 216 = 0. \] 5. Divide the entire equation by 2: \[ x^2 - 3x - 108 = 0. \] 6. Solve the quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 1 \), \( b = -3 \), and \( c = -108 \). Substituting, \[ x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-108)}}{2} = \frac{3 \pm \sqrt{9+432}}{2} = \frac{3 \pm \sqrt{441}}{2}. \] 7. Since \(\sqrt{441} = 21\), \[ x = \frac{3 \pm 21}{2}. \] We discard the negative solution and take \[ x = \frac{3+21}{2} = 12. \] 8. Find the eastern leg: \[ x - 3 = 12 - 3 = 9 \text{ miles}. \] 9. (a) The total distance of the superhighway (alternative) route is the sum of the two legs: \[ 9 + 12 = 21 \text{ miles}. \] 10. (b) Compute the travel times for each route. - Mountain route: \[ \text{Time} = \frac{15 \text{ mi}}{45 \text{ mph}} = \frac{1}{3} \text{ hour} \quad (20 \text{ minutes}). \] - Superhighway route: \[ \text{Time} = \frac{21 \text{ mi}}{60 \text{ mph}} = 0.35 \text{ hour} \quad (\approx 21 \text{ minutes}). \] 11. Since \( \frac{1}{3} \) hour (20 minutes) is less than 0.35 hour (21 minutes), Justin should take the mountain route to minimize the travel time. Final answers: (a) The total distance for the alternative route is \( 21 \) miles. (b) To minimize the trip time, Justin should take the \( 15 \)-mile mountain route.