(1) A particle on the x-axis is moving to the right at 2 units per second. At a certain instant it is at the point (5,0). How rapidly it the distance between the particle and the point (0, 9) on the y-axis changing at that instant?
(2) The length of a rectangle is increasing at the rate of 5 meters per minute while the width is decreasing at the rate of 3 meters per minute. At a certain instant, the length is 20 meters and the width is 10 meters. At that instant determine (a) the rate of change of the area of the rectangle, (b) the rate of change of the perimeter of the rectangle.
(3) A particle on the x-axis is moving to the right at 2 units per second. A second particle is moving down the y-axis at the rate of 3 units per second. At a certain instant the first particle is at the point (5, 0), and the second is a the point (0, 7). How rapidly it the angle between the x-axis and the line joining the two particles changing at that instant? Are the particles moving towards or away from each other at that instant?
(4) The beacon on a lighthouse makes one revolution every 20 seconds. The beacon is 300 feet from the nearest point, P, on a straight shoreline. Find the rate at which the ray of light 60 moves along the shore at a point 200 feet from P. (In other words, if you were driving in a car along the shoreline so that you keep the light constantly on the car, how fast would you be going when you are 200 feet from the point on the shore nearest the light?)
(5) The minute hand of a steeple clock is 4 feet long and the hour hand is 3 feet long. How fast is the distance between the tips of the hands changing at 3 o'clock? (Hints: Use the law of cosines to compute the distance between the tips of the hands. The rates of change of the angle between the hands can be computed since the rate of each hand (radians per minute) is known.)