The table shows a drug absorption
in a typical person.
Notice in the first column, we have the number of hours.
In the second column, we have the amount of drug
in the body in milligrams.
Notice at time 0 there is 1,000 milligrams,
which means 1,000 milligrams of drug was administered.
We're told to let f[t] be the amount of drug
in the body after t hours,
and then we're told to use Desmos
to create a scatter plot of the data.
The next step is to go to desmos.com
and click Graphing Calculator.
To enter the data,
click the plus sign in the upper left-hand corner,
which is the Add Item button,
and then select Table.
We'll enter the time in hours in x1
and the amount of drug in milligrams in y1.
Now the scatter plot has already been created,
we just can't see it because we need to adjust the window,
and there are a couple of ways we can adjust the window.
One way would be to click the minus
in the upper right-hand corner,
which is a Zoom Out button,
but this is not going to work.
Even though we are beginning to see the points,
this is not going to be an ideal window.
We need to click on the wrench
in the upper right-hand corner
and adjust the x- and y-axes manually.
The x-values go from 0 to 24,
and therefore let's change the x-min to -5
and the x-max to 30,
and let's change the step to 4.
I started the x-value at -5
because I do want to see the origin.
And the y-values go from 15 to 1,000.
Let's change the y-minimum to, let's say, -100
and the y-max to 1,100 with a step of 100.
Now if we click on the coordinate plane,
we can see a nice scatter plot for the data,
and we can tell from the scatter plot,
it looks like an exponential decay model
will be a good fit for the data.
Going back to the question,
we're asked to find the exponential function
that fits the data, and we're told
to round to three decimal places as needed,
and our function is going to be f[t].
Going back to Desmos, we go down to cell 2.
We need to use the same variables in the table,
which are x1 and y1.
So to perform exponential regression in the form
of y = a X b to the power of x, we enter y1.
We do not use the equal sign, we use the tilde or twiddle.
So we click Show Keypad, click ABC,
and the twiddle or tilde is to the right
of the curly brackets here.
Then we enter ab raised to the power of x1.
Notice here we have a box that says Log Mode.
We do want to select this in order to use the same method
that the TI-84 and many other software programs use.
Notice right now we have the values of a and b below.
If we click Log Mode though,
they are going to change because we're using a different
method and we do want to use log mode for this question.
We're told to round to three decimal places.
For a we don't have to, a is 1196.2,
and for b we will use 0.843.
So our function is f[t] = 1196.2 X 0.843,
raised to the power of t.
Before we record this, notice how the model
is already graphed on top of the scatter plot.
We can see, we have a very nice model
for the drug absorption data.
So again, our model is f[t]
= 1196.2 X 0.843
raised to the power of t.
Next, based upon the model,
drug is decreasing at what rate per hour?
Give as a percent to one decimal place.
Well, because we have exponential decay,
the base b is equal to 1 - r, where r is the decay rate.
Solving for r, we have r = 1 - b, the base,
which gives us r = 1 - 0.843,
which is equal to 0.157,
which means the drug is decreasing at a rate of
approximately 15.7% per hour.
Next, we're asked to predict the amount of drug
in the body after 30 hours.
We're told to use the graph and round to one decimal place.
So we're going to use the graph to predict
the function value f[30].
I do want to mention that if we use the function
that we just found with the rounded value of b,
we will get a slightly different answer
than using the graph.
But again, we are going to use the graph
to predict this function value.
So going back to Desmos,
we're going to go down to cell 3
and we're going to type in x = 30.
X = 30 is not showing on the coordinate plane,
so let's go ahead and adjust the window again.
We could just click, hold and drag
the coordinate plane to the left,
or we can also go back to the wrench and adjust
the x-maximum, but this looks good.
So now we'll click the point of intersection of the red
vertical line and our black model.
Notice how we can see when t,
or the number of hours, is 30,
the corresponding function value,
or number of milligrams, is approximately 7.1.
After 30 hours, there's approximately
7.1 milligrams of drug still in the body.
Next, we're asked to predict when there will be
250 milligrams of drug in the body.
Again, we are told to round to one decimal place.
So notice here we are given the amount of drug
in the body, or the function value.
We're going to make a prediction for the value of t,
or the time in hours.
So we're trying to make a prediction for t,
given we know f[t] = 250.
Again, we'll go ahead and do this graphically,
going back to Desmos.
Let's go down to cell 4 and enter
f[t] = 250 or just y = 250.
Now we click the point of intersection
of the blue horizontal line and our model.
Notice the function value is 250,
which is the number of milligrams of drug in the body,
and the t value to one decimal place is approximately 9.2.
After approximately 9.2 hours,
there will be 250 milligrams of drug in the body.
Then for the last question,
we're asked to predict the half-life of the drug.
Well remember, 1,000 milligrams was the amount administered.
Since that is the amount of drug in the body at time 0,
half-life is how long it will take
for half that amount to be left, which means we need
to predict the value of t when f[t] = 500.
Again, we'll do this using Desmos.
We will go down to cell 5 and enter f[t], or y = 500.
This time I'll use f[t] just to show
it doesn't matter which method we use.
Now we graph the point of intersection
of our model and the green horizontal line.
Notice how t is approximately 5.1
when the amount of drug in the body is 500 milligrams,
which means the half-life is approximately 5.1 hours.
I hope you found this helpful.