Lesson A ladder foot slides on the ground
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<H2>A ladder foot slides on the ground</H2> <H3>Problem 1</H3>A 13 foot ladder leans against a wall. The foot of a ladder begins to slide away from the wall at the rate of 1 foot per minute. When the foot is 5 ft from the wall, at what rate is the top of the ladder is falling? <B>Solution</B> <pre> Let x be horizontal distance from the wall and y be vertical coordinate. Then from Pythagoras x^2 + y^2 = 13^2. (1) Here x = x(t) and y = y(t) are functions of time, t. Differentiate equation (1) over t. You will get 2x*x'(t) + 2y*y'(t) = 0, hence y't = - (x*x'(t))/y. Evaluate it at the given values x = 5 ft, x'(t) = 1 ft/minute, y = {{{sqrt(13^2 - 5^2)}}} = {{{sqrt(169-25)}}} = {{{sqrt(144)}}} = 12. You will get y'(t) = - {{{(5*1)/12}}} = - 5/12 ft/minute. <U>ANSWER</U>. The top of the ladder moves vertically down at the rate of 5/12 ft per minute. </pre> <H3>Problem 2</H3>A 10 m ladder is leaning against a vertical wall with its foot on the same horizontal plane as the base of the wall. If the lower end of the ladder is moving away from the wall horizontally at 1 m/sec, how fast is the top of the ladder descending when the lower end is 4 meters from the wall? <B>Solution</B> <pre> Let x be horizontal distance from the wall and y be vertical coordinate. Then from Pythagoras x^2 + y^2 = 10^2. (1) Here x = x(t) and y = y(t) are functions of time, t. Differentiate equation (1) over t. You will get 2x*x'(t) + 2y*y'(t) = 0, hence y't = - (x*x'(t))/y. Evaluate it at the given values x = 4 m, x'(t) = 1 m/s, y = {{{sqrt(100 - 4^2)}}} = {{{sqrt(100-16)}}} = {{{sqrt(84)}}}. You will get y'(t) = - {{{(4*1)/sqrt(84)}}} = - 0.436 m/s (rounded). <U>ANSWER</U> </pre> <H3>Problem 3</H3>A 6 m ladder is placed against a wall. If the bottom of the ladder is pushed in towards the wall at 0.4 m/s, how fast is the top of the ladder moving when the ladder reaches 4.5 m above the ground? <B>Solution</B> <pre> Imagine a coordinate plane with coordinate axes x and y (x horizontal, y vertical). Imagine that the bottom of the ladder slides along the horizontal x-axis from some negative x-values to zero (from left to right). Imagine that the top of the ladder slides along the vertical y-axis. The length of the ladder is d = {{{sqrt(x^2 + y^2)}}}. (1) Here x and y are, actually, functions of the time x = x(t), y = y(t). You are given that the derivative of the function x(t) over time t is x'(t) = 0.4 m/s. Square both sides of the formula (1) d^2 = x^2(t) + y^2(t). (2) Since the length of the ladder d is a constant, d^2 is a constant, too, Therefore, the derivative of d^2 over time is zero. From the other side, this derivative, from (2), is 2x*x' + 2y*y'. Thus we have this equation 2x*x' + 2y*y' = 0, or, canceling the factor 2, x*x' + y*y' = 0. (3) When y = 4.5 m above the ground, you have a right angled triangle with the hypotenuse 6 m and vertical leg of 4.5 m. So, its horizontal leg is x = - {{{sqrt(6^2-4.5^2)}}} = -3.969 meters. Now you substitute these data x= -3.969 m, y= 4.5 m, x' = 0.4 m/s into the formula (3). You get then y' = - (x*x')/y = - {{{((-3.969)*0.4)/4.5}}} = 0.353 m/s. <U>ANSWER</U> </pre> My other lessons on Calculus word problems at this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-rate-of-change-of-some-processes.lesson>Finding rate of change of some processes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-defined-by-complicated-expression.lesson>Find the derivative of a function defined by complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Taking-derivative-of-a-function-which-is-defined-implicitly.lesson>Taking derivative of a function, which is defined implicitly</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-satisfying-given-functional-equation.lesson>Find the derivative for a function satisfying given functional equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/09-add-Find-the-Range-of-f%28x%29-=-%285cos%28x%29%29-div-%28x-%2B-1%29%29cos%28x%29.lesson>Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-tricky-Calculus-problem-on-derivative-and-anti-derivative.lesson>A tricky Calculus problem on derivative and anti-derivative</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Couple-of-non-standard-Calculus-problems.lesson>Couple of non-standard Calculus problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-minimum-of-a-function-defined-on-a-curve-in-the-coordinate-plane.lesson>Finding the minimum of a function defined on a curve in a coordinate plane</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Tricky-solution-to-find-the-maximum-of-a-function-defined-by-a-complicated-expression.lesson>Tricky solution to find the maximum of a function defined by a complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximmize-the-volume-of-an-open-box.lesson>Maximize the volume of an open box</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-rectangular-prizm-with-the-given-volume.lesson>Minimize surface area of a rectangular box with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-the-cost-of-an-aquarium-with-the-given-volume.lesson>Minimize the cost of an aquarium with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-conic-paper-cup-with-the-given-volume.lesson>Minimize surface area of a conical paper cup with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-volume-of-a-solid-body-obtained-by-rotation-the-area-about-an-axis.lesson>Find the volume of a solid obtained by rotation of some plane shape about an axis</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-volume-of-a-solid-body-mentally.lesson>Finding the volume of a solid body mentally</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-lessons-on-Calculus-word-problems.lesson>OVERVIEW of my lessons on Calculus word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
