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:
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| 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.
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Solution:
| Step 1:
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| 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
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- 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,}
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| 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.
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| Step 2:
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| Use Implicit Differentiation: We take the derivative of the equation from Step 1 to find
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- 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,}
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| or
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- 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},}
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| Step 3:
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| Evaluate and Solve: At the particular moment we care about,
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- 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.}
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| From this, we can simply plug in to find
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- 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}}
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| 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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| Final Answer:
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| 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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