022 Exam 1 Sample A, Problem 6

A 15-foot ladder is leaning against a house. The base of the ladder is pulled away from the house at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base of the ladder is 9 feet from the house.

Foundations:
Like most geometric word problems, you should start with a picture. This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate.

 Solution:

Step 1:
Write the Basic Equation: From the picture, we can see that the ladder forms a right triangle with the wall and the ground, so we can treat our variables as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2\,\,=\,\,15^2\,\,=\,\,225,}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the height of the ladder on the wall, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is the distance between the wall and the base of the ladder.

Step 2:
Use Implicit Differentiation: We take the derivative of the equation from Step 1 to find
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x\frac{dx}{dt} + 2y\frac{dy}{dt}\,\,=\,\,0,}
or
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt} \,\,=\,\, -\frac{x}{y}\cdot\frac{dx}{dt},}

Step 3:
Evaluate and Solve: At the particular moment we care about,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=9,\quad y=12,\quad dx/dt=2.}
From this, we can simply plug in to find
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt} \,\, = \,\, - \frac {x}{y} \cdot \frac{dx}{dt} \,\, = \,\, -\frac{9}{12} \cdot 2 \,\, = \,\, - \frac{3}{2}}
With units, we have that the ladder is sliding down the wall at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3/2}   feet per second.	Final Answer:
With units, we have that the ladder is sliding down the wall at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3/2}   feet per second.

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