Below is a table with estimates of would population since 1AD

Question

asked Nov 5, 2023 116k views

1 Answer

Lidkxx · Answer 1 · 2023-11-10T16:45:21+0000

Answer:

1)

Step-by-step explanation:

Since we are taking t = 0 as 1900. The initial population is the population in 1900:

We can start writting the exponential model:

:

We know that the population in 1950 (t = 50 in this model) is 2525. Then, we can find k by:

$P(50)=2525=1633e^(k\cdot50)$

And solve for k:

Now, we can apply natural logarithm on both sides:

$\ln((2525)/(1633))=\ln(e^(50k))$

Since natural log and the exponential functions are inverse functions:

$\ln((2525)/(1633))=50k$
$k=(\ln((2525)/(1633)))/(50)$
$k\approx0.0087$

Thus, the exponential model is:

Question 2

Now, we need to use the model we just create to find the predicted population by the model:

At year 1900, t = 0 (that's how the problem asked us to define t)

Then

$P\left(0\right)=1633e^(0.0087\cdot0)=1633e^0=1633$

At year 1920, t = 20

$P(20)=1633e^(0.0087\cdot20)=1633e^(0.174)=1943$

At year 1940, t = 40:

$P(40)=1633e^(0.0087\cdot40)=1633e^(0.348)=2313$

At year 1960, t = 60

$P(60)=1633e^(0.0087\cdot60)=1633e^(0.522)=2752$

At year 1980, t = 80

$P(80)=1633e^(0.0087\cdot80)=1633e^(0.696)=3275$

At year 2000, t = 100

$P(100)=1633e^(0.0087\cdot100)=1633e^(0.87)=3898$

At year 2020, t = 120

$P(120)=1633e^(0.0087\cdot120)=1633e^(1.044)=4639$

At year 2040, t = 140

$P(140)=1633e^(0.0087\cdot140)=1633e^(1.218)=5520$