/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 A boat is pulled into a dock by ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 24

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see figure). (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

Short Answer

Expert verified
The boat's speed is 3ft/s when there is 13 feet of rope out and the winch is retracting rope at the rate of 4ft/s. The winch pulls in rope at 5 ft/s when the boat is moving at a constant rate of 4ft/s with 13 feet of rope out. The speed of the boat decreases, while the speed of the winch increases as the boat gets closer to the dock.

Step by step solution

01

- Express distance in terms of rope length

We can express distance, \(x\), from the boat to the dock as a function of the total amount of rope, \(r\), using the Pythagorean theorem, \(x^2 + 12^2 = r^2\). To find \(x\), we express it as \(x = \sqrt{r^2 - 12^2}\).
02

- Determine the boat speed at 13 feet of rope out

Derive \(x\) with respect to time to find the speed of the boat, which is \(dx/dt\). Using the Chain Rule, \(dx/dt = (1/2)(r^2 - 12^2)^{-1/2} * 2r * dr/dt\). Plugging in \(r = 13ft\) and \(dr/dt = -4ft/s\) (the rate at which the rope is retracted), we find that \(dx/dt = -3ft/s\). This is the opposite of the boat's speed. Therefore, the boat's speed is 3ft/s.
03

- Determine the winch speed at 13 feet of rope out

Now suppose the boat is moving at a constant rate of 4ft/s. Meaning, \(dx/dt = 4ft/s\). Solve for \(dr/dt\), the rate at which the winch pulls in rope, by rearranging the equation \(dx/dt = (1/2)(r^2 - 12^2)^{-1/2} * 2r * dr/dt\) from the previous step to get \(dr/dt = (dx/dt) / ((1/2)(r^2 - 12^2)^{-1/2} * 2r)\). Then plug in the values \(r = 13ft\) and \(dx/dt = 4ft/s\). We find that \(dr/dt = -5ft/s\).
04

- Analyze the boat and winch speed as the boat gets closer to the dock

As the boat gets closer to the dock, \(r\) decreases. Hence, the boat's speed, which is equal to \(-dx/dt\), also decreases. This is because the boat has less distance to cover to reach the dock. On the other hand, the speed at which the winch pulls in rope, represented by \(-dr/dt\), increases. This is because as the boat gets closer to the dock, the winch has to work faster to pull in the slack.

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