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: