Question

A boat is pulled toward a dock by means of a rope wound on a drum that is located 3 ft above the bow of the boat. if the rope is being pulled in at the rate of 4 ft/sec, how fast is the boat approaching the dock when it is 29 ft from the dock? (round your answer to one decimal place.)

Answer 1

Using related rates and the Pythagorean theorem, we can find the rate at which the boat is approaching the dock by differentiating the equation and then substituting the given values to solve for dx/dt.

Step-by-step explanation:

To solve this problem, we need to apply the concepts of related rates in calculus. When the boat is 29 ft from the dock and the rope on the drum is being pulled at 4 ft/sec, we want to find the rate at which the boat approaches the dock. We can represent this using the Pythagorean theorem as a relationship between the distance of the boat from the dock, the length of the rope, and the height at which the rope is attached to the drum.

Let x be the distance between the boat and the dock, y be the length of the rope, and the height from the bow to the drum is a constant 3 ft. So, we have the equation y^2 = x^2 + 3^2, which we can differentiate with respect to time t to find the rate at which x is changing when x = 29 ft. Differentiating both sides, we get 2y*(dy/dt) = 2x*(dx/dt). Plugging in the values, and given dy/dt (the rate at which the rope is being pulled in) is -4 ft/sec, we solve for dx/dt, which gives us the rate at which the boat is approaching the dock.