Final answer:
The student's question involves using the law of cosines to set up an equation involving a right triangle and related rates to find the rate of change of angle θ in radians per hour. This requires calculus and trigonometry skills, typically taught in high school mathematics courses.
Step-by-step explanation:
To determine the rate of change of the angle θ, which is opposite the eastward path of the boat traveling at 42 miles per hour, we can use related rates in trigonometry. Specifically, we can apply the law of cosines in a right triangle scenario to find the angle θ when the boat has traveled 32 miles east.
After the boat has traveled, we have a right triangle with one leg of 29 miles (the initial distance north of Hartville), and another leg of 32 miles (the distance the boat has traveled east). Using the law of cosines:
c² = a² + b² - 2ab cos(θ)
Where:
- a = 29 miles (distance north of Hartville)
- b = 32 miles (distance traveled east by the boat)
- c = distance from Hartville to the boat’s position after 32 miles
Since we are looking for the rate of change of θ, we would differentiate with respect to time. In this case, we have:
d/dt(c²) = d/dt(a² + b² - 2ab cos(θ))
However, since a is constant (29 miles), the differentiation of that term is zero. For θ, we use the chain rule and get:
-2ab sin(θ) (dθ/dt) = 2b(db/dt)
where db/dt is the speed of the boat, which is 42 miles per hour. We can solve for dθ/dt to get the rate of change of θ in radians per hour. However, to solve for dθ/dt, we also need to know sin(θ), which can be calculated using the Pythagorean theorem or found using the initial conditions of a, b, and c. This requires additional steps, which would include solving for c, the hypotenuse of the triangle at the moment the boat is 32 miles east.
The student's question requires the use of trigonometric identities and knowledge of calculus to solve, which usually indicates a high school-level understanding of mathematics. It is an example that applies calculus in the context of trigonometric functions and related rates.