Help with this problem

[randomman10]

Experiments have shown that approximately one acre of farmland per person is needed
to give birth to the earth's population. In 1950, the population of the earth was about
2.5 billion and 1980 amounted to 4.6 billion. If we assume the population
growing exponentially Increases the need for agricultural land exponentially. The earth has
about 9 billion acres of farmland.
From what year does the land not suffice to support the earth's population according to
this model?

[HallsofIvy]

I didn't know farmland "gave birth"to people. I thought women did that! Perhaps this is translation that should have been "supports".

In any case, we are told that the population of earth was 2.5 billion in 1950 and increased "exponentially" to 4.6 billion in 1980. An exponential function is of the form [tex]f(x)= Ca^x[/tex] for some constants, C and a. We need two equations to solve for C and a. Taking x= 0 in 1950 that was [tex]f(0)= C= 2.5[/tex]. For 1980, 30 years later, x= 30 and we have [tex]f(30)= Ca^{30}= 2.5a^{30}= 4.6[/tex]. Divide both sides by 2.5 to get [tex]a^{30}= \frac{4.6}{2.5}= 1.84[/tex]. Taking the 30th root, [tex]a= 1.02[/tex] to 2 decimal places. That is, the earth's population, in billions of people, x years after 1950 is given by [tex]f(x)= 4.6(1.02^x)[/tex].

We are told that one acre of land can support one person and that the earth has 9 billion acres of farm land (and that quantity supposedly does not change). The question is, when will the earth's population reach 9 billion?

What is the value of x so that [tex]4.6(1.02^x)= 9[/tex]? Divide both sides by 4.6 to get [tex]1.02^x= 9/4.6= 1.96[/tex]. Taking the common logarithm of both sides, [tex]log(1.02)x= log(1.96)[/tex] so [tex]x= \frac{log(1.96)}{log(1.02}= \frac{0.29}{0.0086}= 33.72[/tex]. I am going to round that to 34 years since 1950 so 1984.

Oh darn! It looks like we have already starved to death! Well, it was fun while it lasted. (How old is your textbook?)