Calculus question - Need a push in the right direction I'm trying to solve this question:

But I just can't figure out what the first step is, little help? THANKS!
 A: What you have is an initial value problem: a differential equation with a particular value:
$$\frac{df}{dt} = -0.2f,\qquad f(0)=100g.$$
To start, you need to solve for $f$. This is a separable equation, so just separate:
$$\begin{align*}
\frac{df}{dt} &= -0.2f\\
\frac{df}{f} &= -0.2\,dt.
\end{align*}$$
Can you take it from here?
A: If you have not yet been introduced in a formal way to differential equations, look at it this way.
You know that 
$$f'(t) =-0.2f(t)$$
and want to find $f(t)$.
Can you think of a function (other than the identically $0$ function)  whose derivative is (always) $-0.2$ times the function?
Maybe not!  But you have certainly met a function whose derivative is the function itself.
Let $g(t)=e^t$.  Then $g'(t)=e^t$.  More generally, if $g(t)=Ce^t$, where $C$ is a constant, then $g'(t)=g(t)$.
So $e^t$ sort of behaves like you want, except for the $-0.2$ part.  Can you think of a way of modifying $e^t$ so that when you differentiate, the right constant will appear in front?
It would be best to think about this for a while by yourself, but  
Hint:  What happens when you differentiate $e^{kt}$, or more generally $Ce^{kt}$?
A: If $f'(t)$ were equal to $f(t)$ what woud it be? If $f'(t)$ were equal to $-f(t)$ what  woud it be? Your case is a generalization. I describe the standard approach.
Your equation can be rewritten as $y'+0.2y=0$, where $y=f(t)$. This is a homogeneous equation with constant coefficients. The standard method to solve it is by finding the roots of the caracteristic equation. In this case this equation is $X+0.2=0$. Since $X=-0.2$, the general solution is $y=Ae^{-0.2t}$. The initial condition $f(0)=100$ gives $A=100$. Thus $y=f(t)=100e^{-0.2t}$.  
Let's confirm: $y=f(t)=100e^{-0.2t}$, $y'=f'(t)=100(-0.2)e^{-0.2t}=-20e^{-0.2t}$. We do have $-20e^{-0.2t}+0.2\cdot 100e^{-0.2t}\equiv 0$.
A: Your equation can be changed to $f' + 0.2 f = 0$, a homogeneous linear differential equation with constant coefficients. Try to write a polynomial whose roots will help you find a solution to your problem.
