20m away
the dog was 60m away from. you subtract 40m since it is 10m/s x 4 seconds
Answer:
I would say the answer is A... but I'm not so sure ....
Answer:
The rate of change of the area when the bottom of the ladder (denoted by
) is at 36 ft. from the wall is the following:
![\frac{dA}{dt}|_{b=36}=-571.2\, ft^2/s](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%7C_%7Bb%3D36%7D%3D-571.2%5C%2C%20ft%5E2%2Fs)
Explanation:
The Area of the triangle is given by
where
(by using the Pythagoras' Theorem) and
is the length of the base of the triangle or the distance between the bottom of the ladder and the wall.
The area is then
![A=\sqrt{l^2-b^2}b](https://tex.z-dn.net/?f=A%3D%5Csqrt%7Bl%5E2-b%5E2%7Db)
The rate of change of the area is given by its time derivative
![\frac{dA}{dt}=\frac{d}{dt}\left(\sqrt{l^2-b^2}\cdot b\right)](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Cleft%28%5Csqrt%7Bl%5E2-b%5E2%7D%5Ccdot%20b%5Cright%29)
![\implies \frac{dA}{dt}=\frac{d}{dt}\left(\sqrt{l^2-b^2}\right)\cdot b+\frac{db}{dt}\cdot\sqrt{l^2-b^2}](https://tex.z-dn.net/?f=%5Cimplies%20%5Cfrac%7BdA%7D%7Bdt%7D%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Cleft%28%5Csqrt%7Bl%5E2-b%5E2%7D%5Cright%29%5Ccdot%20b%2B%5Cfrac%7Bdb%7D%7Bdt%7D%5Ccdot%5Csqrt%7Bl%5E2-b%5E2%7D)
Product rule
Chain rule
![\implies\frac{dA}{dt}=-\frac{1}{\sqrt{l^2-b^2}}\cdot b^2\cdot \frac{db}{dt}+\sqrt{l^2-b^2}}\cdot \frac{db}{dt}](https://tex.z-dn.net/?f=%5Cimplies%5Cfrac%7BdA%7D%7Bdt%7D%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7Bl%5E2-b%5E2%7D%7D%5Ccdot%20b%5E2%5Ccdot%20%5Cfrac%7Bdb%7D%7Bdt%7D%2B%5Csqrt%7Bl%5E2-b%5E2%7D%7D%5Ccdot%20%5Cfrac%7Bdb%7D%7Bdt%7D)
![\implies\frac{dA}{dt}=\frac{db}{dt}\left(-\frac{1}{\sqrt{l^2-b^2}}\cdot b^2+\sqrt{l^2-b^2}}\right)](https://tex.z-dn.net/?f=%5Cimplies%5Cfrac%7BdA%7D%7Bdt%7D%3D%5Cfrac%7Bdb%7D%7Bdt%7D%5Cleft%28-%5Cfrac%7B1%7D%7B%5Csqrt%7Bl%5E2-b%5E2%7D%7D%5Ccdot%20b%5E2%2B%5Csqrt%7Bl%5E2-b%5E2%7D%7D%5Cright%29)
In here we can identify
,
and
.
The result is then
![\frac{dA}{dt}=8\left(-\frac{1}{\sqrt{39^2-36^2}}\cdot 36^2+\sqrt{39^2-36^2}}\right)=-571.2\, ft^2/s](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%3D8%5Cleft%28-%5Cfrac%7B1%7D%7B%5Csqrt%7B39%5E2-36%5E2%7D%7D%5Ccdot%2036%5E2%2B%5Csqrt%7B39%5E2-36%5E2%7D%7D%5Cright%29%3D-571.2%5C%2C%20ft%5E2%2Fs)