justin must travel from summersville to clayton. he can drive 15 mi through the mountains at 45 mph. or he…

justin must travel from summersville to clayton. he can drive 15 mi through the mountains at 45 mph. or he can drive east and then north on superhighways at 70 mph. the alternative route forms a right angle as shown in the diagram. the eastern leg is 3 mi less than the northern leg.\n(a) find the total distance justin would travel in going the alternative route.\n(b) if justin wants to minimize the time of the trip, which route should he take?

justin must travel from summersville to clayton. he can drive 15 mi through the mountains at 45 mph. or he can drive east and then north on superhighways at 70 mph. the alternative route forms a right angle as shown in the diagram. the eastern leg is 3 mi less than the northern leg.\n(a) find the total distance justin would travel in going the alternative route.\n(b) if justin wants to minimize the time of the trip, which route should he take?

Answer

Explanation:

Step1: Use the Pythagorean theorem

Let the northern leg be (x) mi. Then the eastern leg is ((x - 3)) mi. According to the Pythagorean theorem (a^{2}+b^{2}=c^{2}), where (c = 15) mi (the hypotenuse), (a=x - 3) and (b = x). So ((x-3)^{2}+x^{2}=15^{2}). Expand ((x - 3)^{2}): (x^{2}-6x + 9+x^{2}=225). Combine like - terms: (2x^{2}-6x+9 - 225=0), (2x^{2}-6x - 216 = 0). Divide through by (2): (x^{2}-3x-108=0). Factor the quadratic equation: (x^{2}-3x - 108=(x - 12)(x + 9)=0). Solve for (x): (x=12) or (x=-9). Since distance cannot be negative, (x = 12). The eastern leg is (x-3=12 - 3=9) mi. The total distance of the alternative route is (9 + 12=21) mi.

Step2: Calculate the time for each route

For the mountain route: Use the formula (t=\frac{d}{v}), where (d = 15) mi and (v = 45) mph. So (t_{1}=\frac{15}{45}=\frac{1}{3}) h (=20) min. For the alternative route: The total distance (d = 21) mi and (v = 70) mph. So (t_{2}=\frac{21}{70}=0.3) h (=18) min.

Answer:

a) The total distance of the alternative route is (21) mi. b) Justin should take the alternative (east - then - north) route.