Determining a Function to Model Exponential Growth
The growth of atmospheric carbon dioxide over time can be modeled using a function based on the data from the table below. Now we will determine such a function from the data.
Year | Carbon Dioxide (ppm) |
1990 | 353 |
2000 | 375 |
2075 | 590 |
2175 | 1090 |
2275 | 2000 |
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Find an exponential function that gives the amount of carbon dioxide y in year x.
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Estimate the year when future levels of carbon dioxide will be double the preindustrial level of 280 ppm.
Solutions
- The data points showed exponential growth, so the equation will take the form y=y0ekt. We must find the values of y0 and k. The data began with the year 1990, so to simplify our work we let 1990 correspond to x = 0, 1991 correspond to x = 1, and so on. Since y0 is the initial amount, y0 = 353 in 1990 when x = 0.
Thus the equation is y=353ekx.
From the last pair of values in the table, we know that in 2275 the carbon dioxide level is expected to be 2000 ppm. The year 2275 corresponds to 2275 – 1990 = 285. Substitute 2000 for y and 285 for x and solve for k.y=353ekx
2000=353ek(285)
2000353=e285k
ln2000353=lne285k
ln2000353=285k
k=1285ln2000353≈0.00609
A function modeling the data is y=353e0.00609x
- When the level is double 280 ppm, it will be 2(280)=560 ppm.
y=353e0.00609x
560=353e0.00609x
Divide by 353.
560353=e0.00609x
Take the natural logarithm of both sides.
ln560353=lne0.00609x
Since lnex=x,for all x.
ln560353=0.00609x
x=10.00609⋅ln(560353)≈75.8
Since x = 0 corresponds to 1990, the preindustrial carbon dioxide level will double in the 75th year after 1990, or during 2065, according to this model.