A 17 ft. ladder is leaning against a building. The foot of the ladder is currently 9 ft. from the base of the building and is sliding away from the building at a rate of 3 ft/s. How fast is the top of the ladder sliding down the building?
A 17 ft. ladder is leaning against a building. The foot of the ladder is currently 9 ft. from the base of the building and is sliding away from the building at a rate of 3 ft/s. How fast is the top of the ladder sliding down the building?
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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![**Problem Description for Educational Website**
**Introduction:**
While preparing for your final exam, a classmate expresses difficulty in understanding related rates. To assist, you decide to break down the following problem step by step:
**Problem Statement:**
A 17-foot ladder is leaning against a building. The base of the ladder is currently 9 feet away from the building and is moving away from the wall at a speed of 3 feet per second. Determine the rate at which the top of the ladder is sliding down the building.
**Solution Walkthrough:**
1. **Diagram:**
- Include a right triangle with:
- The ladder as the hypotenuse (17 ft).
- The distance from the building to the bottom of the ladder as one leg (initially 9 ft).
- The height from the ground to the top of the ladder as the other leg.
2. **Identifying Rates:**
- Define variables: Let \( x \) be the distance from the wall to the base of the ladder and \( y \) be the height of the ladder on the wall.
- Given: \(\frac{dx}{dt} = 3 \) ft/s (rate at which the base is moving away).
- Find: \(\frac{dy}{dt} \).
3. **Equation Setup:**
- Using the Pythagorean theorem: \( x^2 + y^2 = 17^2 \).
4. **Solution Explanation:**
- Differentiate the equation with respect to time \( t \):
\[
2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
\]
- Solve for \(\frac{dy}{dt}\).
5. **Interpretation of the Final Answer:**
- Explain the physical meaning of the rate \(\frac{dy}{dt}\) and how it relates to the movement of the ladder.
This walkthrough aims to guide students through solving related rates problems using calculus principles. By following these steps, they will be able to understand the relationship between the position and movement rates of objects in dynamic systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5963b474-a441-468b-8c4e-b34b2e3806b1%2F1c5cf990-c0f4-406f-98b3-7ab5437d169a%2Fz3ycce_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description for Educational Website**
**Introduction:**
While preparing for your final exam, a classmate expresses difficulty in understanding related rates. To assist, you decide to break down the following problem step by step:
**Problem Statement:**
A 17-foot ladder is leaning against a building. The base of the ladder is currently 9 feet away from the building and is moving away from the wall at a speed of 3 feet per second. Determine the rate at which the top of the ladder is sliding down the building.
**Solution Walkthrough:**
1. **Diagram:**
- Include a right triangle with:
- The ladder as the hypotenuse (17 ft).
- The distance from the building to the bottom of the ladder as one leg (initially 9 ft).
- The height from the ground to the top of the ladder as the other leg.
2. **Identifying Rates:**
- Define variables: Let \( x \) be the distance from the wall to the base of the ladder and \( y \) be the height of the ladder on the wall.
- Given: \(\frac{dx}{dt} = 3 \) ft/s (rate at which the base is moving away).
- Find: \(\frac{dy}{dt} \).
3. **Equation Setup:**
- Using the Pythagorean theorem: \( x^2 + y^2 = 17^2 \).
4. **Solution Explanation:**
- Differentiate the equation with respect to time \( t \):
\[
2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
\]
- Solve for \(\frac{dy}{dt}\).
5. **Interpretation of the Final Answer:**
- Explain the physical meaning of the rate \(\frac{dy}{dt}\) and how it relates to the movement of the ladder.
This walkthrough aims to guide students through solving related rates problems using calculus principles. By following these steps, they will be able to understand the relationship between the position and movement rates of objects in dynamic systems.
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