Find the Value - Question 7

Question

In 2000 the population of country A reached 2 ​million, and in 2025 it is projected to be 4.5 million.

​(a) Find values for P0 and a so that the following formula models the population of country A in year x. f(x)=P0ax−2000

​(b) Estimate the​ country's population in 2010 to the nearest hundredth of a million.

​(c) Use f to determine the year during which the​ country's population might reach 7 million.

​(a) Find values for P0 and a.

P0=_____million

a=_____________



​(Round to five decimal places as​ needed.)

Answer 1

So I'm guessing that your function should actually be f(x) = P_0 * a ^(x-2000)

where population f(x) is given in millions and x is the year.

For part (a):

Remember since we are solving for two unknowns we need two equations. We can use the two facts that they gave us to get the two equations that we need.

Fact 1: In 2000 the population of country A reached 2 ​million.

This means that when x = 2000, f(x) = 2.

2 = P_0 * a^(2000-2000) = P_0 * a^0 = P_0

--> 2 = P_0

Fact 2: In 2025 the population is projected to be 4.5 million

This means that for x = 2025, f(2025) = 4.5

4.5 = P_0 * a ^(2025-2000) we already found P_0 = 2 so we can plug that in

4.5 = 2 * a^(2025-2000) = 2* a^(25)

4.5/2 = a^(25)

(4.5/2)^(1/25) = a

---> 1.03297 ≈ a

For part (b):

They are asking what is f(2010)? So x = 2010 and we plug in using our nice new constants found in part (a) to do our calculations.

f(x) = P_0 * a^(x-2000)

f(x) = 2 * 1.03297^(x-2000)

--> f(2010) = 2 * 1.03297^(2010-2000) ≈ 2.77 million people

For part (c):

Now we want to see when the population will reach 7 million. This means that f(x) = 7 and we need to find which x this holds for.

f(x) = 2 * 1.03297^(x-2000) (using the version of our formula with our found constants plugged in)

7 = 2 * 1.03297^(x-2000) now solve for x

7/2 = 1.03297^(x-2000)

ln(7/2) = (x-2000)ln(1.03297)

ln(7/2)/ln(1.03297) = (x-2000)

ln(7/2)/ln(1.03297) + 2000 = x

2038.62 ≈ x BE CAREFUL HERE since it is more than 2038 we have to round up to the next year. Try plugging x = 2038 into the function to see for yourself that it gives a population < 7 million. We need to round up to the nearest integer to get the right answer here.

---> 2039 = x

Plug x = 2039 into f(x) to see for yourself that now we can have a population of 7 million