Math Problem Statement
A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure. The winch pulls in rope at a rate of −0.4 meter per second (ds/dt = −0.4 m/sec) . Find the rate of vertical change and the rate of horizontal change at the end of the pipe when y = 4. (Round your answers to three decimal places.)
Solution
Let's break down the problem step by step.
Step 1: Define the scenario and variables.
We have a winch at the top of a 12-meter building pulling a 12-meter pipe. The pipe is being pulled in by the winch, so the winch reduces the length of the rope at a rate of .
- is the length of the rope being pulled in.
- is the horizontal distance from the bottom of the pipe to the base of the building.
- is the vertical height from the bottom of the pipe to the ground.
Given:
-
The total length of the pipe is 12 meters, so by the Pythagorean theorem: Since the pipe remains 12 meters in length, the equation becomes: Simplifying:
-
The rate at which the rope is being pulled is .
-
We need to find the rate of vertical change () and horizontal change () when .
Step 2: Solve for when .
From the equation , when , we get:
Step 3: Differentiate the equation .
To find the rates of change, differentiate both sides of the equation with respect to time : This simplifies to: x \frac{dx}{dt} + y \frac{dy}{dt} = 0 \tag{1}
Step 4: Relating to and .
Since and , we can differentiate with respect to : Simplifying: Given that and , we have: -0.4 = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{12} \tag{2}
Step 5: Solve for and .
Now, we have two equations:
- Equation (1):
- Equation (2):
Substitute :
From Equation (1), express in terms of : Substitute this into Equation (2): Simplify: Since the terms involving cancel out, we are left with:
Thus, the rate of
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Math Problem Analysis
Mathematical Concepts
Related Rates
Differentiation
Pythagorean Theorem
Formulas
s^2 = x^2 + y^2
2x(dx/dt) + 2y(dy/dt) = 0
ds/dt = (x(dx/dt) + y(dy/dt)) / s
Theorems
Pythagorean Theorem
Chain Rule
Suitable Grade Level
Grades 11-12 (High School Calculus)
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