Question Detail
Question
A boat leaves harbor A heading East at 15 mph. Boat B heads South at 20 mph. Six hours later, boat A is 80 miles out and B is 60 miles out. How fast is the distance \( L \) between them increasing?
Correct Answer
Option A is the correct answer.
Detailed Explanation
( x^2 + y^2 = L^2 \implies 2x\dot{x} + 2y\dot{y} = 2L\dot{L} ). Given ( x=80, \dot{x}=15, y=60, \dot{y}=20 ). ( L = \sqrt{80^2+60^2} = 100 ). So ( 2(80)(15) + 2(60)(20) = 2(100)\dot{L} \implies 2400 + 2400 = 200\dot{L} \implies \dot{L} = 24 \text{ mph} ).
Hint
Use the Pythagorean theorem and differentiate it with respect to time.