Question

The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P0 has doubled in 5 years. Suppose it is known that the population is 12,000 after 3 years. What was the initial population P0? (Round your answer to one decimal place.)

Answer 1

Answer:

The initial population P0 is 7500

Step-by-step explanation:

First of all, we are told that the population is proportional to time. This means that as time goes by, the population will increase. This type of relationship between two variables has the form of a linear equation expressed as:

y = mx + b,

where m is the slope of the curve and b is the intercept of it.

In this case we can state the following equation:

P = mt + P0

Where P is the population at a certain time "t", "m" is the slope of the curve, "t" is the time (this is the independent variable) and P0 is the intercept of the curve and represents the initial population.

Once we state the equation, let's see what we know:

If t = 0, then P = P0

If t = 3, then P = 12000 just as we are told

and if t = 5, then P = 2 P0 because after five years the population has doubled compared to the initial population (P0).

From the definition of the slope of the curve:

(Y2 - Y1)/(X2-X1) = m

and using the data described before we can formulate the slope value:

m = (2P0 - P0)/(5-0)

m = (2P0 - P0)/5

m = P0/5

Then, let's replace the value of m in the following equation:

P = mt + P0

12000 = (P0/5) × 3 + P0 → This is the equation presented when 3 years has gone by and we now have a population of 12000.

12000 = (3/5) P0 + P0

12000 = (8/5) P0

12000×5/8 = P0 = 7500

Then we can calculate the value of the slope m:

m = P0/5

m = 7500/5 = 1500

Now, knowing the value of m and the initial population P0 we are able to calculate the population at any value of "t".