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A 16 foot ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 3 feet per second, how fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 4 feet from the wall? Leave your answer in exact form, simplified as much as possible. Do not include units.

Respuesta :

Answer:

11.625

Explanation:

L = length of the ladder = 16 ft

[tex]v_{y}[/tex] = rate at which top of ladder slides down = - 3 ft/s

[tex]v_{x}[/tex] = rate at which bottom of ladder slides

y = distance of the top of ladder from the ground

x = distance of bottom of ladder from wall = 4 ft

Using Pythagorean theorem

L² = x² + y²

16² = 4² + y²

y = 15.5 ft

Also using Pythagorean theorem

L² = x² + y²

Taking derivative both side relative to "t"

[tex]0 = 2x\frac{\mathrm{d} x}{\mathrm{d} t} + 2y\frac{\mathrm{d} y}{\mathrm{d} t}[/tex]

[tex]0 = x v_{x} + y v_{y}[/tex]

[tex]0 = 4 v_{x} + (15.5) (- 3)[/tex]

[tex]v_{x}[/tex] = 11.625 ft/s

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