1.Try to solve this problem without using a formula. Suppose a disease is transmitted from an individual to one other during the period of a week. Each of those two gives the disease to two more people in the second week, so that the total of newly infected people is now four. Each week the number of infected people double. If the spread of the disease continues, how many weeks will it take a city of 300,000 people to have all been infected?
Use the growth and decay formula to solve the following problem:
Zebra mussels from Europe began invading the Mississippi River in 1988. In 1997 a portion of the river contained an average of 10 zebra mussels per square mile. The exponential growth rate was 340% (k=3.4). Predict the number of mussels per square mile in this portion of the river in 2002 (five years later).
For number 1 I got 19 weeks?
number 2 I am not sure how to do this problem I just need some help with the formula.
Thanks!
Pe^kt
Im not sure what you mean by that??
For the first problem, we can solve it using a recursive approach without using a formula.
Start with the initial infected person. In the first week, this person infects one other individual. In the second week, each of the two infected people infects two more individuals, resulting in a total of four new infections.
If we continue this pattern, we can track the number of new infections each week:
Week 1: 1 infected person, 1 new infection
Week 2: 2 infected people, 2 new infections
Week 3: 4 infected people, 4 new infections
Week 4: 8 infected people, 8 new infections
Week 5: 16 infected people, 16 new infections
We can observe that the number of new infections each week doubles.
To find out how many weeks it will take for a city of 300,000 people to have all been infected, we can continue this pattern until the total number of infected people exceeds 300,000.
Week 6: 32 infected people, 32 new infections
Week 7: 64 infected people, 64 new infections
Week 8: 128 infected people, 128 new infections
Week 9: 256 infected people, 256 new infections
Week 10: 512 infected people, 512 new infections
At Week 10, the number of infected individuals exceeds 300,000. Therefore, it will take 10 weeks for a city of 300,000 people to have all been infected.
Now, let's move on to the second problem with the zebra mussels.
To solve this problem using the growth and decay formula, we need to use the exponential growth formula:
A = P * (1 + r/n)^(n*t)
Where:
A - Final amount
P - Initial amount
r - Growth rate
n - Number of compounding periods per year
t - Time in years
In this case, we are looking for the number of mussels per square mile in 2002, which is 5 years after 1997.
Given that the exponential growth rate is 340% or 3.4 (k = 3.4), and the initial amount is 10 mussels per square mile, we can calculate the final amount:
A = 10 * (1 + 3.4/100)^((1/1)*5)
Simplifying further:
A = 10 * (1 + 0.034)^5
Calculating the value:
A = 10 * (1.034)^5
A ≈ 10 * 1.183
A ≈ 11.83
Therefore, in 2002, there will be approximately 11.83 zebra mussels per square mile in that portion of the river.