If current trends of burning fossil fuels and deforestation continue, then future amounts of atmospheric carbon dioxide in parts per million (ppm) will increase exponentially. In the year of 1990, there was 353 ppm.
A)In the year of 2000, there was 375 ppm of carbon dioxide, find an exponential function that gives the amount of carbon dioxide, y in year t.
B) Estimate the year when future levels of carbon dioxide will double the initial amount of 353 ppm.
Table:
X side is the years:
1)1990
2)2000
3)2075
4)2175
5)2275
Y side is carbon dioxide ppm:
1)353
2)375
3)590
4)1090
5)2000
just using the first two data points, then in year t since 1990, you have
y = 353 e^(kt), so
353e^(10k) = 375
That gives you k, so then you need to find when e^(kt) = 2
A) To find an exponential function that gives the amount of carbon dioxide, y, in a given year, t, we can use the formula for exponential growth:
y = a * e^(bt)
Where:
- y is the amount of carbon dioxide in parts per million (ppm),
- a is the initial amount of carbon dioxide in ppm,
- e is the base of the natural logarithm (approximately 2.71828),
- b is the growth rate of carbon dioxide in ppm per year, and
- t is the number of years since the initial year.
We can use the given data point for the year 2000 to find the values for a and b in our exponential function.
For the year 2000, the amount of carbon dioxide was 375 ppm. Plugging this into the equation, we get:
375 = a * e^(b * 10)
Simplifying this equation, we divide both sides by a:
375/a = e^(b * 10)
To isolate e^(b * 10), take the natural logarithm of both sides:
ln(375/a) = b * 10
Solve for b by dividing both sides by 10:
b = ln(375/a) / 10
Now, we need to find the value of a. We can use the data point for the year 1990, where the amount of carbon dioxide was 353 ppm. Plugging this into our equation, we get:
353 = a * e^(b * 0)
Since any number raised to the power of zero is equal to 1, the equation becomes:
353 = a * 1
Therefore, a = 353.
Now we can substitute the values of a and b into our exponential function:
y = 353 * e^((ln(375/353)/10) * t)
Simplifying further:
y = 353 * e^(0.00913t)
This is the exponential function that gives the amount of carbon dioxide, y, in a given year, t, based on the data points provided.
B) To estimate the year when future levels of carbon dioxide will double the initial amount of 353 ppm, we need to solve for t in the exponential function when y = 706 ppm (double of 353 ppm).
706 = 353 * e^(0.00913t)
Divide both sides by 353:
2 = e^(0.00913t)
To isolate t, take the natural logarithm of both sides:
ln(2) = 0.00913t
Solve for t by dividing both sides by 0.00913:
t = ln(2) / 0.00913
Using a calculator, we find:
t ≈ 75.88
Rounding to the nearest whole number, t ≈ 76.
Therefore, future levels of carbon dioxide will double the initial amount of 353 ppm approximately 76 years after the initial year of 1990.