022 Exam 1 Sample A, Problem 6

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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
${\displaystyle x^{2}+y^{2}\,\,=\,\,15^{2}\,\,=\,\,225,}$
where ${\displaystyle x}$ is the height of the ladder on the wall, and ${\displaystyle y}$ 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
${\displaystyle 2x{\frac {dx}{dt}}+2y{\frac {dy}{dt}}\,\,=\,\,0,}$
or
${\displaystyle {\frac {dy}{dt}}\,\,=\,\,-{\frac {x}{y}}\cdot {\frac {dx}{dt}},}$
Step 3:
Evaluate and Solve: At the particular moment we care about,
${\displaystyle x=9,\quad y=12,\quad dx/dt=2.}$
From this, we can simply plug in to find
${\displaystyle {\frac {dy}{dt}}\,\,=\,\,-{\frac {x}{y}}\cdot {\frac {dx}{dt}}\,\,=\,\,-{\frac {9}{12}}\cdot 2\,\,=\,\,-{\frac {3}{2}}}$
With units, we have that the ladder is sliding down the wall at ${\displaystyle -3/2}$  feet per second.
Final Answer:
With units, we have that the ladder is sliding down the wall at ${\displaystyle -3/2}$  feet per second.