Assume the world population will continue to grow exponentially with a growth constant (corresponding to a doubling time of about 52 years),
it takes acre of land to supply food for one person, and
there are 13,500,000 square miles of arable land in in the world.
How long will it be before the world reaches the maximum population? Note: There were 6.06 billion people in the year 2000 and 1 square mile is 640 acres.
Answer: The maximum population will be reached some time in the year__________?
Hint: Convert .5 acres of land per person (for food) to the number of square miles needed per person. Use this and the number of arable square miles to get the maximum number of people which could exist on Earth. Proceed as you have in previous problems involving exponential growth.
2079
Well, to find the maximum population, first we need to determine the number of people that can be supported by the available arable land.
There are 13,500,000 square miles of arable land in the world, and since 1 square mile is equal to 640 acres, we have a total of 8,640,000,000 acres of arable land.
If it takes 1 acre of land to supply food for one person, then we can say that each person needs 1/640 square miles of arable land for food.
Given that, we can find the maximum number of people that could exist on Earth by dividing the total arable square miles (13,500,000) by 1/640:
Maximum population = 13,500,000 / (1/640)
Maximum population ≈ 8,640,000,000
So, the maximum population the Earth could support is roughly 8.64 billion people.
Now, to find out how long it will take for the world to reach this maximum population, we can use the exponential growth formula.
Since the doubling time is approximately 52 years, we can use the formula:
Final population = Initial population * (2 ^ (time/doubling time))
Given that the population in the year 2000 was 6.06 billion, we can plug in the values to solve for time:
8.64 billion = 6.06 billion * (2 ^ (time/52))
To solve for time, we can take the logarithm (base 2) of both sides:
log2(8.64) = log2(6.06) + (time/52)
time/52 = log2(8.64) - log2(6.06)
time ≈ 52 * (log2(8.64) - log2(6.06))
Calculating this using my handy-dandy calculator... and accounting for the leap year, leap seconds, and everything else, it looks like...
*time passes*
The maximum population will be reached some time in the year 2338, give or take a few clowns.
Disclaimer: Please remember that the calculations made here are simplifications and do not take into account various real-world factors and complexities.
To find the maximum population, we need to calculate the maximum number of people that can be supported by the available arable land.
Given that it takes 1 acre of land to supply food for one person, we can convert this to square miles by using the conversion factor that 1 square mile is equal to 640 acres.
Therefore, 1 person requires 1/640 = 0.0015625 square miles of land for food.
Next, we need to determine the maximum number of people that can be sustained by the available arable land.
The total arable land available is 13,500,000 square miles.
To find the maximum population, we divide the total arable land by the land needed per person:
Maximum Population = 13,500,000 square miles / 0.0015625 square miles per person
Maximum Population = 8,640,000,000 people
Now, let's calculate how long it will take for the population to reach this maximum.
Since the world population is growing exponentially with a growth constant corresponding to a doubling time of 52 years, we can use the exponential growth formula:
P(t) = P0 * e^(kt)
Where:
P(t) is the population at time t
P0 is the initial population at time t=0
k is the growth constant
t is the time in years
Given that in the year 2000 the population was P0 = 6.06 billion (6,060,000,000) people, we can substitute these values into the exponential growth formula and solve for t:
8,640,000,000 = 6,060,000,000 * e^(k*2000)
Rearranging the equation, we get:
e^(k*2000) = 8,640,000,000 / 6,060,000,000
Taking the natural logarithm of both sides, we can isolate the exponent:
k*2000 = ln(8,640,000,000 / 6,060,000,000)
k = ln(8,640,000,000 / 6,060,000,000) / 2000
Now, we can use this value of k to determine the time it will take for the population to reach the maximum:
t = ln(8,640,000,000 / 6,060,000,000) / k
Calculating this using a calculator, we find that t is approximately 30.3 years.
Therefore, the maximum population will be reached in approximately 30.3 years from the year 2000.
To find out how long it will be before the world reaches the maximum population, we need to follow these steps:
Step 1: Convert the .5 acres of land per person to the number of square miles needed per person.
Since 1 square mile is equal to 640 acres, we can calculate the number of square miles needed per person by dividing .5 acres by 640. This gives us:
.5 acres / 640 acres = 0.00078125 square miles per person
Step 2: Calculate the maximum number of people who could exist on Earth based on the available arable land.
To do this, we multiply the number of arable square miles (13,500,000 square miles) by the number of square miles needed per person (0.00078125 square miles per person). This gives us:
13,500,000 square miles * 0.00078125 square miles per person = 10,546,875,000 people
So, the maximum number of people who could exist on Earth is approximately 10,546,875,000.
Step 3: Use the growth constant to determine the doubling time.
The growth constant given in the question corresponds to a doubling time of about 52 years. This means that the population will double every 52 years.
Step 4: Calculate how many doubling periods are needed to reach the maximum population.
To find out how many doubling periods are needed, we can divide the required population (10,546,875,000) by the current population (6.06 billion or 6,060,000,000). This gives us:
10,546,875,000 / 6,060,000,000 = 1.7394 doubling periods
Since we can't have a fraction of a doubling period, we can round this up to the nearest whole number. Therefore, it will take 2 doubling periods to reach the maximum population.
Step 5: Calculate the time it will take to reach the maximum population.
We multiply the doubling time (52 years) by the number of doubling periods. In this case, the doubling period is 2, so:
52 years * 2 doubling periods = 104 years
Therefore, it will take approximately 104 years for the world to reach the maximum population.
Note: The year in which the maximum population will be reached will depend on the starting point. In this case, we are given the year 2000 as the starting point, so the maximum population will be reached in the year 2104.