Question

Total emissions of carbon dioxide from the burning of fossil fuels have been increasing at about

3​%

per year​ (data from 2010 to​ 2011). If emissions continue to increase at this​ rate, about how much higher will total emissions be in

2040

than in​ 2010?

Using the approximate​ formula, emissions will increase by a factor of ?

between 2010 and

2040

Using the exact​ formula, emissions will increase by a factor of

?

between 2010 and

2040

Answer 1

If emission increase by 3% every year, then emission after t years will be

(1.03)^t original emissions.

Putting t=30 in a calculator, we get 2.42

So, by 2040 using exact formula, emissions increase by a factor of 2.42.

We can use the approximation of (1+x)^t = 1 + x*t

This formula is more accurate as x gets smaller. Since 0.03 is not that small, we don't expect the approximation to be very close.

taking x = 0.03 and t = 30

we get

(1.03)^30 = 1.9. So by the approximate formula emission increases by a factor of 1.9.