Asked by Silvia Gopalakrishnan on Apr 27, 2024
Verified
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=Cekty = C e ^ { k t }y=Cekt for the population. (Let t=0t = 0t=0 correspond to the year 2000.)
A) y=19.5e0.0045ty = 19.5 e ^ { 0.0045 t }y=19.5e0.0045t
B) y=7.2e1.2076ty = 7.2 e ^ { 1.2076 t }y=7.2e1.2076t
C) y=19.5e0.2076ty = 19.5 e ^ { 0.2076 t }y=19.5e0.2076t
D) y=19.5e0.0104ty = 19.5 e ^ { 0.0104 t }y=19.5e0.0104t
E) y=7.2e0.0604ty = 7.2 e ^ { 0.0604 t }y=7.2e0.0604t
Growth Model
A mathematical representation used to describe how a quantity changes over time.
Population
The whole number of people or inhabitants in a country or region.
- Master the concept of exponential amplification and diminution in a range of settings, like population expansion and radioactive deterioration.
Verified Answer
IE
Isabella ElaineApr 27, 2024
Final Answer :
D
Explanation :
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 }
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 }
Learning Objectives
- Master the concept of exponential amplification and diminution in a range of settings, like population expansion and radioactive deterioration.
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