If the population is 2 million and is increasing at 4% each year. and the annual food supply an supply 4 million but the the population rate goes up .5 each year. in how many years will there not be enough food

Solution

We can first express the country's population, p(t), in millions of people, as a function of the time t, measured in years from the initial time. Since we know the initial population p(0)=2 and the annual growth rate is 4%, then p(t) is an exponential function:
p(t)=2(1.04)t.

We are also given that the food supply grows at a constant rate. So we can express the country's food supply at time t, which we call f(t), as a linear function of t. Again, we know the initial value f(0)=4 and the constant rate of change is 0.5 million people per year, so we have:

f(t)=4+.5t.

Population_and_food__0b8e68db3f6589fe24c3ee35a51b9030
We are looking for the value of t which makes p(t) greater than f(t) for the first time.

We see from the graph that the two functions intersect at around t=78. So after 78 years the food supply is just barely enough for the country's population. After this point, however, we see that p(t)>f(t), so this country will first experience shortages of food after approximately 78 years.
If the country doubled its initial food supply, our new function for the food supply would be
h(t)=8+.5t
We would expect food shortages to occur, if at all, later than in part (a).
Population_and_food__733f1f7e58ef40bc21366ee85cd699d7
Again looking at the graph, we see that the two functions intersect, and so food shortages would still occur. We find p(t)=h(t) at roughly t=81. So, the country will first experience food shortages after 81 years. So doubling the initial food supply delays the eventual food shortage by only 3 years.
If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, we have the new food supply function:
j(t)=8+t
We would expect, in this case, for food shortages to occur much later than in part (b), if at all.
Population_and_food__c8963d902bc506f5a01a9084ed9a7445
Looking at the graph we see that this time the food shortage occurs at t=103, about 25 years later than in part (a).

Examining the behavior of the exponential function more closely we observe, that the slope of the exponential function keeps increasing whereas the slope of any linear function is constant. Even if a linear function has a very large slope, an exponential function will eventually grow even faster and overtake the linear function.

I hope that helps!