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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{2}+y^{2}\,\,=\,\,15^{2}\,\,=\,\,225,}
where $x$ 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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