The temperature of Earth s surface is strongly influenced by the amount of carbon in the atmosphere. Each year, human activity adds another 9 gigatons of carbon to the atmosphere. 2 gigatons of this carbon is absorbed by the oceans, and 3 gigatons of this carbon is absorbed by plant growth (photosynthesis). There is currently 900 gigatons of carbon in the atmosphere.

a) Assume that these are the only flows of carbon into/out of the atmosphere, and that the given rates stay constant over time. Write down a differential equation for the total amount of carbon in the atmosphere over time. Solve the equation and use the solution to predict how much carbon there will be in the atmosphere in 25 years.

(b) Let us make our a bit more model realistic. Instead of assuming a constant rate of absorption by plant growth, assume that the rate of carbon absorption by plant growth is proportional to the amount of carbon in the atmosphere at any time t. This increase constitutes 0.4% for each gigaton of carbon. Write down a differential equation for the total amount of carbon in the atmosphere over time. Solve the equation and use the solution to predict the equilibrium amount of carbon in the atmosphere.

(c) Now expand the model again and imagine that humans start to reduce their carbon emissions such that the rate at which carbon is added to the atmosphere decreases over time. Assume that this decreasing rate is described by the function h(t) = 7e^-at +2; where a > 0 is a constant.

d) Again write out and solve the differential equation describing the amount of carbon in the atmosphere. Assuming that a = 0.002, calculate how long it will take for the amount of atmospheric carbon to reach it maximum value. What is this maximum value?