Interpreting change in exponential models: changing units | High School Math | Khan Academy

The amount of carbon dioxide (CO2) in the atmosphere increases rapidly as we continue to rely on fossil fuels. The relationship between the elapsed time T in decades—let me highlight that because that's not a typical unit—but in decades since CO2 levels were first measured and the total amount of CO2 in the atmosphere. So, the amount of CO2 A of D sub T in parts per million is modeled by the following function.

So, the amount of CO2 as a function of how many decades have passed is going to be this. So, T is in decades in this model right over here. Complete the following sentence about the yearly rate of change. The yearly rate of change in the amount of CO2 in the atmosphere, round your answer to two decimal places.

Every year, the amount of CO2 in the atmosphere increases by a factor of... If they said every decade, well this would be pretty straightforward. Every decade you increase T by one, and so you're going to multiply by 1.06 again. So, every decade you increase by a factor of 1.06. But what about every year?

I always find it helpful to make a bit of a table just so we can really digest things properly. So, I'll say T and I'll say A of T. So when T is equal to zero—so at the beginning of our study—well, 1.06 to the zero power is just going to be one. You have 3155 parts per million.

So, what's a year later? A year later is going to be a tenth of a decade—remember T is in decades—so a year later is 0.1 of a decade. So 0.1 of a decade later, what is going to be the amount of carbon we have? Well, it's going to be 3155 times 1.06 to the 0.1 power. And what is that going to be? Well, let's see.

If we calculate it, 1.06 to the 0.1 power is equal to approximately 1.58. So, this is the same thing as 3155 * 1.58, and I should say approximately equal to... I did a little bit of rounding there. So after another year—now we're at T equals 0.2, we're at 0.2 of a decade—where are we going to be?

We're going to be at 3155 * 1.06 to the 0.2, which is the same thing as 3155 * (1.06 to the 0.1) raised to the 2 power. So we're going to multiply by this 1.06 to the 1/10 power again, or we're going to multiply by 1.58 a second time.

Another way to think about it, if we want to reformulate this model in terms of years, for each year of T, it's going to be 3155. Now, our common ratio wouldn't be 1.06; it'd be 1.06 to the 0.1 power, or approximately 1.58. Then we would raise that; now T would be in years.

Now, here it is in decades, and I could say approximately since this is rounded a little bit. So every year, the amount of CO2 in the atmosphere increases by a factor of... I could say 1.06 to the 0.1 power. But if I'm rounding my answer to two decimal places, well, we're going to increase by 1.58. In fact, they should—they increase by a factor of... I'm guessing they want more than two decimal places. Well, anyway, this right over here is five significant digits, but I'll leave it there.