Explanation Given: Alina props a 12-foot ladder against the side of her house. Concept Used: Pythagorean Theorem: In a right angle triangle, the sum of the squares of legs of triangle is equal to the square of the hypotenuse of the right triangle. Calculation: Letx be the distance of the base of the ladder from the house wall and y be the distance of the top of the ladder from the ground as shown below: Since the base of the ladder is sliding away at a rate of half of a foot per second, so d x d t = 1 2 foot per second Observe that the ladder is making a right triangle with the wall of house and the ground. So, using the Pythagorean Theorem in the right triangle, it gives x 2 + y 2 = ( 12 ) 2 x 2 + y 2 = 144 ...... ( i ) Now, to find the rate at which the top of ladder moving down the side of house, differentiate above equation ( i ) implicitly with respect to time t and simplify further as shown below: d d t ( x 2 + y 2 ) = d d t ( 144 ) d d t ( x 2 ) + d d t ( y 2 ) = 0 2 x d x d t + 2 y d y d t = 0 2 y d y d t = − 2 x d x d t d y d t = − 2 x 2 y d x d t d y d t = − x y ⋅ d x d t ...... ( i i ) Now, at the moment when the top of the ladder is 6 feet from the ground, the length of the base of ladder from the house can be found by substituting y = 6 in ( i ) as shown below x 2 + y 2 = 144 x 2 + ( 6 ) 2 = 144 x 2 + 36 = 144 x 2 = 108 x = ± 108 x = 108

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 3.5, Problem 51A

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The rate at which the area of the triangle formed by ladder, house and ground changing when the top of ladder is 6 feet from the ground.

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Chapter 3.5, Problem 50A

Chapter 3.5, Problem 52A

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The rate at which the area of the triangle formed by ladder, house and ground changing when the top of ladder is 6 feet from the ground.

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The rate at which the area of the triangle formed by ladder, house and ground changing when the top of ladder is 6 feet from the ground.

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