**Problem Statement:** The top end of a ladder slides down a vertical wall as its bottom end slides across a horizontal floor. At a certain instant, the speed of the top end is \( k \) times the speed of the bottom end. Find the acute angle (i.e., between \( 0 \) and \( \frac{\pi}{2} \)) that the ladder makes with the ground at this instant.**Detailed Explanation:**This problem involves understanding the relationship between the rates of change of the top and bottom ends of a sliding ladder and finding a specific angle at which this relationship holds true. We are given that the speed of the top end is \( k \) times the speed of the bottom end and asked to determine the acute angle that the ladder makes with the horizontal floor at this specific moment.This scenario can be analyzed with the principles of related rates in calculus. Suppose the length of the ladder is constant and denoted as \( L \). Let \( x(t) \) be the distance of the bottom end of the ladder from the wall at time \( t \), and let \( y(t) \) be the distance of the top end of the ladder from the ground at time \( t \). By the Pythagorean theorem,\[x(t)^2 + y(t)^2 = L^2\]Differentiating both sides with respect to time \( t \), we get\[2x(t) \frac{dx(t)}{dt} + 2y(t) \frac{dy(t)}{dt} = 0\]Simplifying, we find\[x(t) \frac{dx(t)}{dt} + y(t) \frac{dy(t)}{dt} = 0\]Given that the speed of the top end is \( k \) times the speed of the bottom end, we have\[\left| \frac{dy(t)}{dt} \right| = k \left| \frac{dx(t)}{dt} \right|\]Substituting this into our differentiated equation:\[x(t) \left| \frac{dx(t)}{dt} \right| + y(t) \left( k \left| \frac{dx(t)}{dt} \right| \right) = 0\]Solving for the ratio of \( x(t) \) and

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The top end of a ladder slides down a vertical wall as its bottom end slides across a horizontal floor. At a certain instant, the speed of the top end is k times the speed of the bottom end. Find the acute angle (ie. between 0 and 1/2) that the ladder makes with the ground at this instant.

The top end of a ladder slides down a vertical wall as its bottom end slides across a horizontal floor. At a certain instant, the speed of the top end is k times the speed of the bottom end. Find the acute angle (ie. between 0 and 1/2) that the ladder makes with the ground at this instant.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter3: Linear And Nonlinear Functions

Section: Chapter Questions

Problem 26MCQ

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