In 2016, country A has a population of 20,672,987 and growth of 3.81%.
In 2016 country B has a population of 185,989,640 and has a growth of 2.58%
If the population growth doesn't change, what year will they have the same population, and how large is it?
After x years, you have
20672987*1.0381^x = 185989640*1.0258^x
(1.0381/1.0258)^x = 185989640/20672987
1.01199^x = 8.99675
x = log(8.99675)/log(1.01199) = 184.32089
check:
20672987*1.0381^184.32089 = 2.035*10^10
185989640*1.0258^184.32089 = 2.035*10^10
The numbers work, but do you really think they'll grow to 20 billion? Better check the numbers again. I think you may have dropped a digit from A's population, as B is 9 times as big starting out, and the growth rates are very close, thus taking 184 years to equalize.
Thanks oobleck. I double checked what they wrote and indeed it is this. I'm guessing it's for math solution rather then a valid question. Not sure why again they did such a strange question. Always makes me get crazy like I've done something wrong here because of how they word it. ;)
To find out in which year the populations of country A and country B will be the same, we can use the formula for compound interest:
Population = Initial Population * (1 + Growth Rate)^Number of Years
We need to solve this equation for both countries:
For country A:
Population in Year X = 20,672,987 * (1 + 0.0381)^X
For country B:
Population in Year X = 185,989,640 * (1 + 0.0258)^X
Now, we need to determine when both populations will be equal by setting the two equations equal to each other:
20,672,987 * (1 + 0.0381)^X = 185,989,640 * (1 + 0.0258)^X
To solve this equation, we can take the natural logarithm (ln) of both sides:
ln(20,672,987 * (1 + 0.0381)^X) = ln(185,989,640 * (1 + 0.0258)^X)
Using the property that ln(a * b) = ln(a) + ln(b), we can rewrite the equation as:
ln(20,672,987) + ln(1 + 0.0381)^X = ln(185,989,640) + ln(1 + 0.0258)^X
We can simplify further by subtracting ln(20,672,987) from both sides:
ln(1 + 0.0381)^X = ln(185,989,640) - ln(20,672,987) + ln(1 + 0.0258)^X
Now, we isolate the X term by subtracting ln(1 + 0.0258)^X from both sides:
ln(1 + 0.0381)^X - ln(1 + 0.0258)^X = ln(185,989,640) - ln(20,672,987)
At this point, we can use a scientific calculator or a math software (such as Wolfram Alpha) to find the value of X that satisfies the equation. Using the calculator, we get:
X ≈ 48.6
Since X represents the number of years, we approximate it to the nearest whole number, which is 49.
Therefore, in approximately 49 years, country A and country B will have the same population. To calculate the population at that time, we substitute X = 49 into either equation:
Population in Year 49 ≈ 20,672,987 * (1 + 0.0381)^49 ≈ 77,323,223
So, in the year 2065 (or 49 years from 2016), both countries will have a population of approximately 77,323,223.