The population of a region for 2000 is 19.5 (in millions). The predicted population (in millions)for the region in the year 2020 is 24. Find the constants C (to one decimal place)and k (to four decimal places)to obtain the experimental growth model $y = C e ^ { k t }$ for the population. (Let $t = 0$ correspond to the year 2000.)
A) $y = 19.5 e ^ { 0.0045 t }$
B) $y = 7.2 e ^ { 1.2076 t }$
C) $y = 19.5 e ^ { 0.2076 t }$
D) $y = 19.5 e ^ { 0.0104 t }$
E) $y = 7.2 e ^ { 0.0604 t }$

Question

Quizplus · Accepted Answer

We can use the given data to solve for C and k.

When t = 0 (corresponding to the year 2000), y = 19.5. 
This gives us:

19.5 = C e ^ { k 0 } 
19.5 = C

So we know that C = 19.5.

Now we can use the second piece of information to solve for k.

When t = 20 (corresponding to the year 2020), y = 24. 
This gives us:

24 = 19.5 e ^ { k 20 }

Dividing both sides by 19.5:

1.230769231 = e ^ { k 20 }

Taking the natural logarithm of both sides:

ln(1.230769231) = ln(e ^ { k 20 })

ln(1.230769231) = k 20

k = ln(1.230769231)/20

k ≈ 0.0104 (rounded to four decimal places)

Therefore, the experimental growth model is:

y = 19.5 e ^ { 0.0104 t }

The Population of a Region for 2000 Is 19 y=Cekty = C e ^ { k t }y=Cekt

The Population of a Region for 2000 Is 19 $y = C e ^ { k t }$