Most population questions ask about when the population will reach etc. This one's different though. The second one is with cones but I'm not sure if the volume is increasing per unit of time or height.
I'll give you some hints, but I'm sure janvdl will give you the exact process.
For problem 1:
We want: , but as you see, is not a function of t.
Here's how we take care of that.
Find: at . I hope that makes sense to you.
For problem 2:
We want: when .
Use the fact that:
Thus:
, where to solve this.
Good luck!
Let's use the hint of the great ecMathGeek
Someone please check my computation, I caught myself making a lot of mistakes while doing this, I don't know if i caught all my mistakes.
We want , which is the rate of change of with respect to time . We will use the chain rule to obtain this. we can find by the following operation:
since derivative notations can function as fractions, the would cancel, leaving , which is what we want to find.
So let's find
when ,
Now let's find :
......by the Chain Rule
Now, when ,
So when :
So now, finally, we find :
When :
...weird number