Could someone help me with this? I strictly want to be guided, I don't want the answer to be told to me.

Question

lifeisadangerousgame · Answer

attachment

solomonzelman · Answer

$x$ is the distance between spotlight and the man.
$y$ is the height of the shadow.
$12$ is the distance from the spotlight to the wall.
$2$ is the man's height.|dw:1480274279284:dw|

solomonzelman · Answer

So, you know that $\color{black}{\displaystyle \cot \theta = x/2=12/y }$

lifeisadangerousgame · Answer

How did you know to use cot?

solomonzelman · Answer

I can use tangent, or I can just note these triangles are similar ... either way I will lead to the same relation, $\color{black}{\displaystyle xy=24 }$

lifeisadangerousgame · Answer

Like, why are we using cot/tan? What will it give us?

solomonzelman · Answer

if we use tangent is the ratio of opp/adj, and I know that $\tan \theta$ is equal in the inner triangle and in the entire (big) triangle.

For the inner triangle I have $\color{black}{\displaystyle \tan \theta =2/x }$, and for the big triangle we have $\color{black}{\displaystyle \tan \theta =y/12 }$. This allows us to set $\color{black}{\displaystyle 2/x =y/12 }$ which leads to $\color{black}{\displaystyle xy=24 }$.

solomonzelman · Answer

Same way we can use $\cot \theta$, to set
$\color{black}{\displaystyle \cot \theta =x/2 }$  and  $\color{black}{\displaystyle \cot \theta =12/y }$, which then allows 
$\color{black}{\displaystyle x/2=12/y\quad \Longleftrightarrow \quad  }$ $\color{black}{\displaystyle xy=24 }$.

solomonzelman · Answer

Or, again, you can just note the triangles are similar.
The key thing here is that the ratios $\color{black}{\displaystyle 2/x  }$ and $\color{black}{\displaystyle y/12  }$ are equivalent,
(or that he ratios $\color{black}{\displaystyle x/2  }$ and $\color{black}{\displaystyle 12/y  }$ are equivalent).

lifeisadangerousgame · Answer

Okay, I see, that makes sense now, I just didn't get why we were doing that

solomonzelman · Answer

Alrighty ...

solomonzelman · Answer

So, now we have $\color{black}{\displaystyle x(t)y(t)=24  }$.

solomonzelman · Answer

Can you differentiate that wrt t ?

lifeisadangerousgame · Answer

dx/dt * dy/dt = 0?

solomonzelman · Answer

Yup.

solomonzelman · Answer

Recall, that you are looking for the "how fast is the length of his shadow, y, on the wall is decreasing, when he is 4m away from the building".

phi · Answer

***dx/dt * dy/dt = 0?****

I think you want to use the product rule. You  will get something different from this

solomonzelman · Answer

Oh, yes, true ... how didn't I notice that, ty!

solomonzelman · Answer

In general, $\color{black}{\displaystyle \frac{d}{dt}[f(t)g(t)]=f'(t)g(t)+f(t)g(t)  }$, right?
(We are undoing the differentiation, it was incorrect)