Final answer:
Using the concept of vectors and the Law of Cosines, the distance between two cars traveling at 60 mi/hr and 80 mi/hr on paths that differ by 66° for 40 minutes is found to be approximately 66.4 miles when rounded to one decimal place.
Step-by-step explanation:
To solve this problem, we can use the concept of vectors and their magnitudes. Since the two cars travel at angles that differ by 66° and their speeds are given, we can calculate the distance between them after a certain time using the Law of Cosines, which is applicable in non-right triangle scenarios. First, we need to convert the time from minutes to hours because the speeds are in miles per hour. There are 60 minutes in an hour, so 40 minutes is ⅓ of an hour (or approximately 0.667 hours).
The distances traveled by the two cars after 40 minutes can be calculated by multiplying their speeds by the time in hours:
Car 1 (60 mi/hr) × 0.667 hr = 40 miles,
Car 2 (80 mi/hr) × 0.667 hr = 53.36 miles.
Now, applying the Law of Cosines to the triangle formed by the paths of the two cars, we get:
Distance apart 2 = Car 1 distance 2 + Car 2 distance 2 - 2 × Car 1 distance × Car 2 distance × cos(66°). Plugging the values we get:
Distance apart 2 = 402 + 53.362 - 2 × 40 × 53.36 × cos(66°). Before calculating, recall that cos(66°) is approximately 0.4067. Solving this using a calculator gives the distance apart as approximately 66.4 miles, rounded to one decimal place.