Two pilots take off from the same airport. Mason heads due south. Nancy heads 23º west of south. After 400 land miles, how far is Nancy from Mason’s route? Round your answer to the nearest tenth of a mile. (Enter only the number.)
To find out how far Nancy is from Mason's route, we can break down the problem into two components: the east-west component and the north-south component.
First, let's calculate the north-south component for Nancy's route. We know that Mason heads due south, so his north-south component is 400 miles.
Now, let's calculate Nancy's north-south component. Nancy heads 23º west of south, which means her angle with respect to the south direction is (180º - 23º) = 157º. Using trigonometry, we can calculate the north-south component for Nancy's route:
North-south component = 400 miles * sin(157º)
Using a calculator, we find that the north-south component for Nancy's route is approximately -381.7 miles (rounded to the nearest tenth).
Since the north-south component is negative, it means Nancy is 381.7 miles south of Mason's route.
Now, let's calculate the east-west component for Nancy's route. Since Nancy is heading 23º west of south, her east-west component will be:
East-west component = 400 miles * cos(157º)
Using a calculator, we find that the east-west component for Nancy's route is approximately -163.0 miles (rounded to the nearest tenth).
Since the east-west component is negative, it means Nancy is 163.0 miles west of Mason's route.
To find out the distance between Nancy and Mason's route, we can use the Pythagorean theorem:
Distance = √(north-south component^2 + east-west component^2)
Plugging in the values we found:
Distance = √((-381.7)^2 + (-163.0)^2)
Using a calculator, we find that the distance between Nancy and Mason's route is approximately 419.4 miles (rounded to the nearest tenth).
Therefore, Nancy is approximately 419.4 miles from Mason's route.