A 26 foot long ladder is leaning against a vertical wall. The foot of the ladder is 10 feet away from the base of the wall. The foot of the ladder is being pulled away from the base of the wall at a rate of 4 feet per second. How fast is the top of the ladder sliding down the wall at this instant? (related rates)

- anonymous

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- myininaya

related rates problem awesome!

- anonymous

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- myininaya

cyter beat me

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- myininaya

lol wrong problem

- myininaya

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- anonymous

could both answers be correct?

- myininaya

lol there are two different answers
let me see if i made a mistake

- anonymous

You used 4 instead of 10

- myininaya

cyter is right

- myininaya

i used y' for y my bad

- myininaya

cyter wins

- anonymous

we both win because we spent energy working on the problem

- myininaya

lol

- anonymous

in the writing in the purple, isn't the derivative of s^2 2s, not just 2?

- anonymous

yep

- anonymous

the right hand side should have been 2s ds/dt
I knew something was amiss.

- myininaya

since s is a constant (it doesnt change)
s'=0
so 2ss'=0

- myininaya

so it doesnt change the answer

- anonymous

yep it makes no difference but it looks logical

- myininaya

right
and it is best to be logical during to look crazy

- myininaya

than not during*

- anonymous

but what happens to the 2s? does it cancel out? and how does the x (dx/dt) becomes negative?

- myininaya

2ss'=0

- myininaya

the derivative of s^2 is 2ss'
but since s doesnt change s'=0

- anonymous

oo right, sorry

- anonymous

factor out the 2 s and devide both sides by 2
0/2 = 0

- myininaya

cyter did you draw that on the computer? it looks very mechanical like

- myininaya

on some parts

- myininaya

lol looks better than my horrible hand writing

- anonymous

\[\text{2((xdx/dt)+(ydy/dt) = 2sds = 0/dt}\] dividing both sides by 2 we have
\[\text{((xdx/dt)+(ydy/dt) = sds/dt = 0}\] now subtract both sides by (xdx/dt) we get
\[\text{(ydy/dt) = -(xdx/dt)}\] and then divide both sides by y to get dy/dt alone
\[\text{(dy/dt) = -(x/y)dx/dt}\] Then plug and play

- anonymous

yes I used paint which isn't very good

- anonymous

Your handwriting is legible, unlike mine where I have to use a drawing program on my computer

- myininaya

i have a scanner haha

- myininaya

lol

- anonymous

ok i get it now, thank you both of you for your help

- anonymous

glad to help

- anonymous

Using cyter's solution diagram,
\[y=x \text{Tan}[\text{ArcCos}[x/26]] \]
then
\[y=26 \sqrt{1-\frac{x^2}{676}} \]
The total derivative of the above is:
\[\text{Dt}[y]==-\frac{x \text{Dt}[x]}{26 \sqrt{1-\frac{x^2}{676}}} \]
Replace x with 10 and Dt[x] with 4 and simplify,
\[\text{Dt}[y]==-\frac{5}{3} \]
I hope there are no errors. First DE solved in years.

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