## Ladder sliding down wall problem (Implicit Diff)

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### Ladder sliding down wall problem (Implicit Diff)

A 10 ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate
of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet
from the wall?

Mark, Agent...somebody would like help setting this one up. I am sure this is another Implicit application. Until I get comfortable with these... I am going to keep bugging out every time one pops up
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mathkid
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### Re: Ladder sliding down wall problem (Implicit Diff)

I am concerned that you keep asking the same type of questions and don't seem to have learned anything from the responses you get. Draw a picture pf the ladder, ground and house. Do you see a right triangle? Label the two legs of the triangle and use the Pythagorean theorem to write a formula involving those two lengths. Differentiate both sides to get an equation involving the rates of change of those. You are told the distance from the house to the base of the ladder and its rate of change. You can use those two equations, the Pythagorean theorem and its derivative, to find the height of the ladder on the house and $$its$$ rate of change.
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### Re: Ladder sliding down wall problem (Implicit Diff)

ok
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mathkid
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### Re: Ladder sliding down wall problem (Implicit Diff)

mathkid wrote:A 10 ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate
of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet
from the wall?

Mark, Agent...somebody would like help setting this one up. I am sure this is another Implicit application. Until I get comfortable with these... I am going to keep bugging out every time one pops up

You are correct, it is another implicit differentiation. The ladder, wall, and floor make a right triangle so pythagorean theorem applies. DRAW THE PICTURE, with c as hypotenuse and sides x on floor, y on wall.

$x^2 + y^2 = c^2$

$\frac{d}{dt} \ (x^2+ y^2) \ = \ \frac{d}{dt} \ (c^2)$

$2x \ \frac{dx}{dt} \ + \ 2y \ \frac{dy}{dt} \ = 2c \ \frac{dc}{dt}$

divide out 2 then plug in everything you know and we'll continue from there...

you should be able to get x, dx/dt, y, c, and dc/dt using the info given

that means dy/dt can be solved for.

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agentredlum
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### Re: Ladder sliding down wall problem (Implicit Diff)

so is this right ..?

b^2 + h^2 = 10^2
h^2 = 100- b^2
h = root 100 - b^2

now I know from the past that Area = 1/2 base times height

db/dt = 1

b = 6

so A = 1/2 [ b (root 100-b^2]

now go dA/dT = 1/2[dB/dt(100-b^2)^1/2 + B (1/2) (100-b^2)^-1/2 (you said take out sq root agent)

dA/dT = 1/2 [dB/dT (root 100-b^2) - B/root 100-B^2 times dB/dT

dA/dT = 1/2[8-4.5]

dA/dT = 1.75 ft^2/sec

?
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mathkid
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### Re: Ladder sliding down wall problem (Implicit Diff)

You're not looking for dA/dt, so area formula is not usefull. You are looking for ROC of side against the wall.

y^2 = 10^2 - 6^2

y^2 = 64

y = 8

so, so far you have x = 6 , y = 8 , c = 10 , dx/dt = 1

if you find dc/dt then you will have 5 of the 6 unknowns in ...

$x \ \frac{dx}{dt} \ + \ y \ \frac{dy}{dt} \ = \ c \ \frac{dc}{dt}$

what is the ROC of hypotenuse, dc/dt = ?

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agentredlum
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### Re: Ladder sliding down wall problem (Implicit Diff)

can u give me a hint?
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mathkid
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### Re: Ladder sliding down wall problem (Implicit Diff)

as the ladder slides down, does the length of the hypotenuse change?
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agentredlum
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### Re: Ladder sliding down wall problem (Implicit Diff)

since the ladder is the hypotanuse I don't see how it's lengh can change it seems to me the lengh of the ladder is fixed
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### Re: Ladder sliding down wall problem (Implicit Diff)

correct! very good,

now, if c is the hypotenuse AND IT'S CONSTANT AT 10, dc/dt = ?
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agentredlum
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### Re: Ladder sliding down wall problem (Implicit Diff)

0 ?
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mathkid
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### Re: Ladder sliding down wall problem (Implicit Diff)

Bravo!

Now plug all the known values into

$x \ \frac{dx}{dt} + y \ \frac{dy}{dt} = c \frac{dc}{dt}$

and show me what you get.

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agentredlum
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### Re: Ladder sliding down wall problem (Implicit Diff)

I get..

6(1) + 8 (dont have value for dy/dt ??) = 10(0)
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mathkid
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### Re: Ladder sliding down wall problem (Implicit Diff)

Yes, very good, you are going to solve for dy/dt,

let me see you do that.

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agentredlum
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### Re: Ladder sliding down wall problem (Implicit Diff)

6 + 8dy/dt = 0

8dy/dt = -6

dy/dt= -6/8

dy/dt = -3/4

is this right agent?
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