Sec (30) (08300 8. You lean a 20 foot ladder is leaning against a wall so you can replace a burned out light at the top of the wall. You push the bottom of the ladder toward the wall at a rate of 1 foot per second to raise the top of the ladder along the wall toward the light. When the base of the ladder is 12 ft from the wall, how fast is the top of the ladder moving up the wall? √x² + y² = 1² Ⓒ 2x / 42 + 2y o/x = 1² ot dy 2x = x + 2y = 1 = 0 ot de df 2+ = 147/5 dt Page 4 of 7 20ft, 12 ft
Sec (30) (08300 8. You lean a 20 foot ladder is leaning against a wall so you can replace a burned out light at the top of the wall. You push the bottom of the ladder toward the wall at a rate of 1 foot per second to raise the top of the ladder along the wall toward the light. When the base of the ladder is 12 ft from the wall, how fast is the top of the ladder moving up the wall? √x² + y² = 1² Ⓒ 2x / 42 + 2y o/x = 1² ot dy 2x = x + 2y = 1 = 0 ot de df 2+ = 147/5 dt Page 4 of 7 20ft, 12 ft
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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Question
![**Problem Statement:**
8. You lean a 20-foot ladder against a wall so you can replace a burned-out light at the top of the wall. You push the bottom of the ladder toward the wall at a rate of 1 foot per second to raise the top of the ladder along the wall toward the light. When the base of the ladder is 12 ft from the wall, how fast is the top of the ladder moving up the wall?
**Solution Steps:**
1. \( x^2 + y^2 = L^2 \)
Here, \( L = 20 \) feet is the length of the ladder, \( x \) is the distance of the base of the ladder from the wall, and \( y \) is the height of the ladder on the wall.
2. Differentiate the equation with respect to time \( t \):
\[
2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
\]
3. Given:
\(\frac{dx}{dt} = 1 \text{ ft/s}\)
When \( x = 12 \text{ ft} \), find \(\frac{dy}{dt}\).
4. Solve for \(\frac{dy}{dt}\):
\[
2(12)(1) + 2y \frac{dy}{dt} = 0
\]
\[
24 + 2y \frac{dy}{dt} = 0
\]
\[
y \frac{dy}{dt} = -12
\]
\[
\frac{dy}{dt} = \frac{-12}{y}
\]
**Diagram Explanation:**
The diagram on the right shows a right triangle formed by the ladder, the wall, and the ground. It includes a hypotenuse labeled as 20 ft (the ladder), a base labeled as 12 ft (the distance from the wall), and an unlabeled height along the wall.
**Page Reference:**
Page 4 of 7](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5670fba-02c8-4d09-8c4d-0ebec62c9518%2F3a23df4d-53bb-426e-812e-426a3ccda019%2F8fyadaz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
8. You lean a 20-foot ladder against a wall so you can replace a burned-out light at the top of the wall. You push the bottom of the ladder toward the wall at a rate of 1 foot per second to raise the top of the ladder along the wall toward the light. When the base of the ladder is 12 ft from the wall, how fast is the top of the ladder moving up the wall?
**Solution Steps:**
1. \( x^2 + y^2 = L^2 \)
Here, \( L = 20 \) feet is the length of the ladder, \( x \) is the distance of the base of the ladder from the wall, and \( y \) is the height of the ladder on the wall.
2. Differentiate the equation with respect to time \( t \):
\[
2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
\]
3. Given:
\(\frac{dx}{dt} = 1 \text{ ft/s}\)
When \( x = 12 \text{ ft} \), find \(\frac{dy}{dt}\).
4. Solve for \(\frac{dy}{dt}\):
\[
2(12)(1) + 2y \frac{dy}{dt} = 0
\]
\[
24 + 2y \frac{dy}{dt} = 0
\]
\[
y \frac{dy}{dt} = -12
\]
\[
\frac{dy}{dt} = \frac{-12}{y}
\]
**Diagram Explanation:**
The diagram on the right shows a right triangle formed by the ladder, the wall, and the ground. It includes a hypotenuse labeled as 20 ft (the ladder), a base labeled as 12 ft (the distance from the wall), and an unlabeled height along the wall.
**Page Reference:**
Page 4 of 7
Expert Solution
Step 1
Given query is to find the rate of change of the top of the ladder.
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