Country A has a growth rate of 2.1% per year. The population is currently 5,747,000 and the land area of Country A is 36,000,000,000 square yards. Assuming this growth rate continues and is exponential, after how long will there be one person for every square yard of land.This will happen in what years?
To find out after how long there will be one person for every square yard of land in Country A, we first need to calculate the total population when this condition is met.
Given that the land area of Country A is 36,000,000,000 square yards, our goal is to reach a population that matches this land area. The population density can be calculated by dividing the population by the land area:
Population density = Population / Land area
Population density = 5,747,000 / 36,000,000,000
Next, we need to determine the time it takes for the population to grow to match the population density.
To calculate the growth rate over a certain time period, we can use the formula:
Population at time t = Population at time 0 * (1 + growth rate)^t
Where:
Population at time 0 = Initial population = 5,747,000
Growth rate = 2.1% per year = 0.021
Population at time t = Population density = 5,747,000 / 36,000,000,000
Substituting the values into the formula, we have:
5,747,000 / 36,000,000,000 = 5,747,000 * (1 + 0.021)^t
Solving for t, we can rearrange the equation:
(1 + 0.021)^t = 5,747,000 / 36,000,000,000
Now, we can take the logarithm of both sides to solve for t:
t * log(1.021) = log(5,747,000 / 36,000,000,000)
t = log(5,747,000 / 36,000,000,000) / log(1.021)
Using a calculator to solve the equation, we find that t ≈ 542.224
Therefore, it will take approximately 542.224 years for there to be one person for every square yard of land in Country A.
To find out after how long there will be one person for every square yard of land, we need to calculate the future population and compare it to the land area.
First, let's find the future population using the exponential growth formula:
P = P0 * e^(rt)
Where:
P = Future population
P0 = Initial population
r = Growth rate (in decimal form)
t = Time (in years)
e = Euler's number (approximately 2.71828)
In this case:
P0 = 5,747,000 (current population)
r = 2.1% or 0.021 (growth rate in decimal form)
Let's calculate P for a specific time period, starting at t = 0:
P = 5,747,000 * e^(0.021t)
Next, we need to determine when the future population will equal the land area, which is 36,000,000,000 square yards.
5,747,000 * e^(0.021t) = 36,000,000,000
To solve this equation for t, we need to isolate the exponential term. Divide both sides of the equation by 5,747,000:
e^(0.021t) = 36,000,000,000 / 5,747,000
Now, take the natural logarithm (ln) of both sides to eliminate the exponential:
ln(e^(0.021t)) = ln(36,000,000,000 / 5,747,000)
Using the logarithmic property that ln(e^x) = x, we can simplify:
0.021t = ln(36,000,000,000 / 5,747,000)
Finally, divide both sides by 0.021 to solve for t:
t = ln(36,000,000,000 / 5,747,000) / 0.021
Using a calculator, you can find the value of t. The answer will represent the time in years after which there will be approximately one person for every square yard of land.
Note: Since this growth rate assumes exponential growth, it's important to note that it might not be a realistic projection for such a long timespan. Population growth rates can vary, and factors like resource availability, technological advancements, and societal changes can influence the actual growth rate.
5,747 * 1.021^n = 36,000,000
1.021^n = 36,000,000/5,747 =
n = log (36,000,000 / 5,747)/log 1.021
= 421
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check
1.021^421 = 6307
6307*5,747,000 = 3.6*10^10
= 3.6*10,000,000,000 =36,000,000,000
sure enough