In the previous section, we found one point that we knew had to be on the best-fitting curve. If we could find another point on this curve, we could then find the curve since we would have two unknowns and two equations. However, we do not know one for sure, therefore we have to guess. What would be a good guess for another point? How about the y-intercept?
Next, we find the equation of the curve that goes through these two points. Recall the exponential equation is in this form,
We have two variables, a and b, to find.
The spreadsheet can be set up to do this for us. In cell E7, the spreadsheet calculated the first guess for b. Lets look at how.
The two equations are found by substituting the mean data points as x and y for one equation and substituting the y intercept for the other. Notice that the y intercept has an x value of 0. Since e^b*0 is 0. (Why?), this equation reduces to 8.3 = a. Therefore, we can plug in this value for a in the other equation and solve for b.
Thus, we have our first guess at the best fitting curve. In the spreadsheet, place this value of b you get in cell E7 in cell E9. We can then find a, by plugging in our values for b, x and y. Remember, that x and y have to be the mean data points since we know this point to be on the curve.
If all went correctly, you should have found this equation for your first guess.
Take a look at the plot of this equation and the original data points.
What do you think? It looks like it does a pretty good job for smaller values of x, but what about larger values? Do you think we can find a better fitting curve? Remember, that we only know for sure that one of the points is on the best-fitting curve. We assumed the y-intercept could be on the curve, but we do not know this for sure! In the next section we will explore a statistical way to find how well this first equation fits the data.
1.) Solve this equation for a and b.
a e^(b * 2) = 10
a e^(b * 0) = 25
2.) Here is the Polonium plot again.
Find one equation that fits this data. Use the polonium spreadsheet.
3.) From the equation you found in question 3. How much of the element should there be at time 50 days? at time 200 days? How close are these predictions to the actual data points?
4.) If someone asked you to predict how much of the element there would be at time 500 days, how confident would be in your answer?