**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

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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