Answer:
a) 
b) See the proof below

c) 
Explanation:
Part a
For this case we have the following differential equation:

With the initial condition 
We can rewrite the differential equation like this:

And if we integrate both sides we got:
Where
is a constant. If we apply exponential for both sides we got:

Using the initial condition
we got:

So then our solution for the differential equation is given by:

For the half life we know that we need to find the value of t for where we have
if we use this condition we have:


Applying natural log we have this:

And then the value of t would be:

And using the fact that
we have this:

Part b
For this case we need to show that the solution on part a can be written as:

For this case we have the following model:

If we replace the value of k obtained from part a we got:


And we can rewrite this expression like this:

And we can cancel the exponential with the natural log and we have this:

Part c
For this case we want to find the value of t when we have remaining 
So we can use the following equation:

Simplifying we got:

We can apply natural log on both sides and we got:

And if we solve for t we got:

We can rewrite this expression like this:

Using properties of natural logs we got:
