A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac{π}{4}$ $pi/4$ rad?

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A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac{π}{4}$ $pi/4$ rad?

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Answer 1 · 2018-04-06T16:18:35

$\frac{\sqrt{2}}{5}$ $sqrt2/5$ $r a d$ $rad$ / $s$ $s$

Explanation:

Let $x$ $x$ be the distance between the base of the wall and the bottom of the ladder. And let $θ$ $theta$ be the angle between the top of the ladder and the wall,

Then $\frac{x}{10} = sin θ$ $x/10 = sin theta$ , so $x = 10 sin θ$ $x = 10sintheta$ .

Differentiating with respect to time $t$ $t$ gets us

$\frac{d x}{d t} = 10 cos θ \frac{d θ}{d t}$ $dx/dt = 10 cos theta (d theta)/dt$

We were told that $\frac{d x}{d t} = 2$ $dx/dt = 2$ $f t$ $ft$ / $s$ $s$ and we seek $\frac{d θ}{d t}$ $(d theta)/dt$ when $θ = \frac{π}{4}$ $theta = pi/4$

$2 = 10 (cos (\frac{π}{4})) \frac{d θ}{d t}$ $2 = 10 ( cos(pi/4)) (d theta)/dt$

$2 = 10 (\frac{\sqrt{2}}{2}) \frac{d θ}{d t} = 5 \sqrt{2} \frac{d θ}{d t}$ $2 = 10 (sqrt2/2) (d theta)/dt = 5sqrt2 (d theta)/dt$

$\frac{d θ}{d t} = \frac{2}{5 \sqrt{2}} = \frac{\sqrt{2}}{5}$ $(d theta)/dt = 2/(5sqrt2) = sqrt2/5$ $r a d$ $rad$ / $s$ $s$

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A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac{π}{4}$ $pi/4$ rad?

1 Answer

Explanation:

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A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is π4pi/4 rad?

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Explanation:

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A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac{π}{4}$ $pi/4$ rad?