Calculus 1: Related Rates Problems

Let $y = \sin{(x)}$

Find $\frac{dy}{dt}$ when $x = \frac{\pi}{4}$ given $\frac{dx}{dt} = 2 \frac{cm}{sec}$

A boat is pulled in by means of a winch on a dock 12 ft above the deck of the boat. If the winch pulls in rope at the rate of 4 ft/sec, determine the speed of the boat when 13 feet of rope is out.

A man 6 ft tall is walking away from a streetlight 20ft high at a rate of 5ft/sec. At what rate is the tip of his shadow moving when he is 24 feet from the lightpost and at what rate is the length of his shadow increasing?

During a night run, an observer is standing 80 feet away from a long, straight fence when she notices a runner running along it, getting closer to her. She points her flashlight at him and keeps it on him as he runs. When the distance between her and the runner is 100 feet, he is running at 9 feet per second. At this moment, at what rate is she turning the flashlight to keep him illuminated? Include units in your answer.

An airplane is flying at an altitude of 7 miles and passes directly over a radar antenna as shown in the figure. When the plane is 10 miles from the antenna, the radar detects that the distance between the plane and the tower is changing at the rate of 300 mph. What is the speed of the airplane at that moment?