In this video, we will determine
an exponential decay function in the form of P of t
equals a times b raised to the power of t.
We will also not be using logarithms.
A radioactive substance decays exponentially.
A scientist begins with 160 milligrams
of a radioactive substance.
After 25 hours, 120 milligrams of the substance remains.
How many milligrams will remain after 32 hours?
Round to a 10th of a milligram.
Remember, when we have an exponential function in the form
of P of t equals a times b raised to the power of t,
a is the initial or starting amount.
In this case, because the scientist begins
with 160 milligrams of a radioactive substance,
we know a, the initial or starting amount, is equal to 160.
And therefore, our function must be in the form of P of t
equals 160
times b raised to the power of t.
We also know, after 25 hours,
120 milligrams of the substance remains.
As an ordered pair in the form of t comma P of t,
the ordered pair would be 25 comma 120.
Using this ordered pair, we now substitute 25 for t
and 120 for P of t in order to solve for the base b.
Performing this substitution gives us 120 equals 160
times b raised to the power of 25.
And now to solve for b, we first isolate b
by dividing both sides by 160.
And now simplifying,
on the right, 160 divided by itself simplifies to one,
giving us b to the 25th power on the right side.
On the left, 120 over 160 simplifies to 3/4.
The greatest common factor between 120 and 160 is 40.
There are three 40s in 120
and four 40s in 160.
And now to solve for b, we need to undo the 25th power
by raising both sides to the reciprocal power of 25,
which is 1/25.
So we'll raise both sides of the equation to the 1/25 power.
Notice now on the right side,
we have powers raised to powers,
and therefore we multiply the exponents.
25 times 1/25 is equal to one,
giving us b to the first, or b, on the right side.
On the left side, we have 3/4
raised to the power of the 1/25.
Now we're going to leave b in its exact form.
We will not convert to a decimal and round.
Because if we round,
we will have more of an error in our exponential function.
So now we'll substitute b back in to our function P of t,
which will give us P of t
equals 160.
Now because the base b has a rational exponent,
let's put the base b in brackets.
So we have open bracket
and then 3/4 raised to the power of 1/25,
and this base b is still raised to the power of t.
Now let's go ahead and simplify this.
Notice how here we have powers raised to powers,
and therefore we multiply the exponents.
1/25 times t is 1/25t, or t divided by 25,
and therefore our simplified
exponential decay function is P of t
equals 160 times 3/4
raised to the power of t divided by 25.
Now that we have the exponential decay function,
we can determine how much remains after 32 hours
by determining P of 32.
P of 32 is equal to 160 times 3/4
raised to the power of 32 divided by 25.
And now we'll go to the calculator
and round to the nearest 10th of a milligram.
To the nearest 10th, we have 110.7 milligrams.
Let's write the answer as a complete sentence.
We now know, after 32 hours,
110.7 milligrams of the substance remains.
I hope you found this helpful.