A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat (see figure). The winch pulls the rope at a rate of 4 feet per second. Find the speed of the boat when 15 feet of rope is out. What happens to the speed of the boat as it gets closer and closer to the dock? STEP 1: STEP 2: STEP 4: 2x Let x be the distance from the boat to the dock and y be the length of the rope. Write the original equation. 12² + x² = Differentiate the equation with respect to x and y. dx dt dx 99 99 는 dx dt dt 12 dt dx Not draws to scale STEP 3: Solve for dx/dt (in ft/sec) by substituting the values for x, y, and dy/dt. (Round your answer for dx/dt to one decimal place.) 15 -(-4) 114 ft/sec = = = = ft/sec dy dt dy dt ---Select--- State the appropriate conclusion. As x → 0, dx dt 8
A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat (see figure). The winch pulls the rope at a rate of 4 feet per second. Find the speed of the boat when 15 feet of rope is out. What happens to the speed of the boat as it gets closer and closer to the dock? STEP 1: STEP 2: STEP 4: 2x Let x be the distance from the boat to the dock and y be the length of the rope. Write the original equation. 12² + x² = Differentiate the equation with respect to x and y. dx dt dx 99 99 는 dx dt dt 12 dt dx Not draws to scale STEP 3: Solve for dx/dt (in ft/sec) by substituting the values for x, y, and dy/dt. (Round your answer for dx/dt to one decimal place.) 15 -(-4) 114 ft/sec = = = = ft/sec dy dt dy dt ---Select--- State the appropriate conclusion. As x → 0, dx dt 8
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding the Speed of a Boat Being Pulled by a Winch
A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat. The winch pulls the rope at a rate of 4 feet per second. We need to find the speed of the boat when the rope is 15 feet long, and understand what happens to the speed as the boat gets closer to the dock.
#### Diagram Explanation
The diagram shows a right triangle formed by the winch at the top of the dock, the horizontal line representing the boat's position, and the rope being the hypotenuse.
### Steps in the Problem
#### STEP 1
Let \( x \) be the distance from the boat to the dock, and \( y \) be the length of the rope. The original equation, based on Pythagoras' Theorem, is:
\[ 12^2 + x^2 = y^2 \]
#### STEP 2
Differentiate the equation with respect to \( x \) and \( y \):
\[ 2x \frac{dx}{dt} = 2y \frac{dy}{dt} \]
Solve for:
\[ \frac{dx}{dt} = \frac{y}{x} \frac{dy}{dt} \]
#### STEP 3
Substitute the given values to solve for \( \frac{dx}{dt} \) (the speed of the boat):
Given:
- \( y = 15 \)
- \( \frac{dy}{dt} = -4 \) (since the rope is being pulled in)
- Solve for \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = \frac{15}{x}(-4) \]
The value of \( x \) needs to be calculated using the equation from STEP 1:
\[ 12^2 + x^2 = 15^2 \]
\[ 144 + x^2 = 225 \]
\[ x^2 = 81 \]
\[ x = 9 \]
Plugging in:
\[ \frac{dx}{dt} = \frac{15}{9}(-4) \]
\[ \frac{dx}{dt} = \frac{2}{3}(-4) \]
\[ \frac{dx}{dt} = -\frac{8}{3} \approx -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3578f3e4-1bc3-4b4b-9f3b-a9b3ea5dbfae%2F219a1622-76f5-49c6-8c0e-34746f9d3922%2F013ya7c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Speed of a Boat Being Pulled by a Winch
A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat. The winch pulls the rope at a rate of 4 feet per second. We need to find the speed of the boat when the rope is 15 feet long, and understand what happens to the speed as the boat gets closer to the dock.
#### Diagram Explanation
The diagram shows a right triangle formed by the winch at the top of the dock, the horizontal line representing the boat's position, and the rope being the hypotenuse.
### Steps in the Problem
#### STEP 1
Let \( x \) be the distance from the boat to the dock, and \( y \) be the length of the rope. The original equation, based on Pythagoras' Theorem, is:
\[ 12^2 + x^2 = y^2 \]
#### STEP 2
Differentiate the equation with respect to \( x \) and \( y \):
\[ 2x \frac{dx}{dt} = 2y \frac{dy}{dt} \]
Solve for:
\[ \frac{dx}{dt} = \frac{y}{x} \frac{dy}{dt} \]
#### STEP 3
Substitute the given values to solve for \( \frac{dx}{dt} \) (the speed of the boat):
Given:
- \( y = 15 \)
- \( \frac{dy}{dt} = -4 \) (since the rope is being pulled in)
- Solve for \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = \frac{15}{x}(-4) \]
The value of \( x \) needs to be calculated using the equation from STEP 1:
\[ 12^2 + x^2 = 15^2 \]
\[ 144 + x^2 = 225 \]
\[ x^2 = 81 \]
\[ x = 9 \]
Plugging in:
\[ \frac{dx}{dt} = \frac{15}{9}(-4) \]
\[ \frac{dx}{dt} = \frac{2}{3}(-4) \]
\[ \frac{dx}{dt} = -\frac{8}{3} \approx -
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