population density HELP

Question

Population and consumption - On January 1st, 2017, the world population was approximately 7.362 × 109 . At this time the population was estimated to grow at 1.05% per year and all evidence suggests it will stay at this rate. At the same time there is about 7.68 billion acres of arable land on the earth and each year it is being reduced by 24 million acres due the effects of climate change, deforestation, and human expansion. 1) Create a model to predict the population of the planet as a function of time. Use your model to predict the world population in 2050. 2) Create a function to predict the acres of arable land as a function of time. Use your model to predict the acres of arable land in 2050. 3) According to your models what is ratio of arable land per person in the year 2050? By current consumption rates the average person on planet earth uses 1 acre of arable land per year to sustain themselves. If the ratio of acres of arable land per person dips below 1 we find ourselves without enough food to feed the world population. 4) In what year will the ratio of acres of arable land per person be one? What happens after that? 5) The situation is not this overwhelmingly bleak (although it is not exactly sunshine and rainbows either). What do you think mathematics can tell us about this problem beyond just when we’ll have to face it? Remember mathematics is very powerful and doesn’t know when to quit

Answer 1

1) Growing by a fixed percent (1.05%) every year is exponential growth.

P(t)= 7.362·1.0105^t

where P(t) is the population year t, 7.362 is the starting population (in 2017) in billions of people, 1.05 is the growth rate, and t is the number of years since 2017. In 2050, t = 2050 - 2017 = 33. Plug t = 33 into P(t) to compute the estimated population in 2050. Use your calculator.

2) Changing by the same amount (24 million) every year is linear growth or decay:

A(t) = 7.68 - (0.024)t

where A(t) is the number of arable acres in year t, 7.68 is the initial amount of arable acres (in 2017), 0.024 is the number of acres lost per year (converted from millions to billions of acres), and t is the number of years since 2017. Again in 2050 t = 2050 - 2017 = 33 years. Plus t = 33 in to A(t) to compute the estimated number of arable acres in 2050. Use your calculator.

3) Take the ratio P/A for 2050 using what you computed in 1) and 2).

4) Set P(t) = A(t) and solve for t. You will have to use logs to get the t out of the exponent. That is the year the P/A ratio will equal 1. This can be tricky so maybe it's better to just graph P(t) and A(t) on a graphing calculator or app and find the intersection.

5) After the year t you get in step 4, there will be less than one acre of arable land per person. That means that there is exactly one acre of farmable land to feed you. After this time the ratio will continue to decline.

This whole exercise is a modified form of a calculation first done in the early 1800s by the economist Thomas Malthus, and is known as the Malthusian Doctrine. It states that population grows faster (exponentially) than the means we have to feed it (arable or farmable land). The end result is that we will eventually be unable to feed ourselves if the population continues to increase exponentially.