PLEASE HELP WITH THIS-ITS DUE APRIL 5TH, 2011!!!
During a past epidemic, the number of infected people increased exponentially with time, or f (t) = ab^t , where f is the number of infected people at time t, in days. Suppose that at the onset of the epidemic (0 days) there are 30 people infected, and 6 days after the onset there are 300 people infected.
a) Specify the function f, that models this situation, by determining a and b.
b) Predict the number of people who will be infected at the end of 2 weeks using your answer to a.
c) According to this function, approximately when will the number of infected people reach 10,000? (Use logarithms to sove this part - a guess and check answer WILL NOT be accepted)
F(t) = 30b^6 = 300,
30b^6 = 300,
Divide both sides by 30:
b^6 = 10,
Take log of both sides:
6log(b) = log10,
log(b) = log10 / 6,
log(b) = 1/6,
Exponential form:
10^(1/6) = b,
Or b = 10^(1/6).
F(t) = 30*10^(1/6)^t,
a. Eq: F(t) = 30*10^(t/6).
b. F(14)=30*10^(14/6) = 6463 Infected.
c. 30*10^(t/6) = 10000,
Divide both sides by 30:
10^(t/6) = 10000/30.
(t/6)log10 = log10000 - log30 = 2.5229.
(t/6)log10 = 2.5229.
Divide both sides by log10:
t/6 = 2.5229,
Multiply both sides by 6:
t = 6 * 2.5229 = 15.14 = approximately 15 Days.
a) To determine the function f, we need to find the values of a and b. We are given two data points: at the onset of the epidemic (0 days), there are 30 infected people, and 6 days later, there are 300 infected people.
Let's start by plugging in the first data point into the exponential function: f(0) = ab^0 = 30. Since any number raised to the power of 0 is 1, this simplifies to a = 30.
Now, let's plug in the second data point: f(6) = ab^6 = 300. We know that a = 30, so we can substitute that in: 30b^6 = 300.
To solve for b, we can divide both sides of the equation by 30: b^6 = 10. Taking the 6th root of both sides gives b ≈ 1.5178.
Therefore, the function f that models this situation is f(t) = 30(1.5178)^t.
b) To predict the number of people who will be infected at the end of 2 weeks, we need to find the value of f(t) where t is 14 (since there are 7 days in a week).
Plugging in 14 into the function f(t) = 30(1.5178)^t gives f(14) ≈ 30(1.5178)^14 ≈ 5,488.58.
Therefore, we can predict that approximately 5,489 people will be infected at the end of 2 weeks.
c) To determine when the number of infected people will reach 10,000, we need to find the value of t for which f(t) = 10,000.
Using the function f(t) = 30(1.5178)^t, we can set it equal to 10,000:
30(1.5178)^t = 10,000.
To solve this equation using logarithms, we can take the logarithm of both sides (base doesn't matter):
log(30(1.5178)^t) = log(10,000).
Using properties of logarithms, we can simplify this equation to:
log(30) + t * log(1.5178) = log(10,000).
Now, isolate t by subtracting log(30) from both sides:
t * log(1.5178) = log(10,000) - log(30).
Finally, divide both sides by log(1.5178) to solve for t:
t = (log(10,000) - log(30)) / log(1.5178).
Using a calculator, we can determine that t ≈ 21.69.
Therefore, according to this function, the number of infected people will reach 10,000 approximately 21.69 days after the onset of the epidemic.