di/dt = k i
di/i = k dt
ln i = kt + c
i = e^(kt+c) = C e^kt
here
200/7 = k (1000)
k = .02875
i = 1000 e^(.02875 t)
Write and exponential growth model for the epidemic. Let t represent time in days.
I got k=.0260459367 but then what's the fianl equation?
di/i = k dt
ln i = kt + c
i = e^(kt+c) = C e^kt
here
200/7 = k (1000)
k = .02875
i = 1000 e^(.02875 t)
i = 1000 e^kt
1200 = 1000 e^7k
e^7k = 1.2
7 k = ln 1.2
k = .0260459367 as you said
N(t) = N0 * e^(kt)
where N(t) represents the number of people infected at time t, N0 represents the initial number of infected people when the epidemic is discovered, and k represents the growth rate constant.
Given that 1,000 people are infected when the epidemic is first discovered (t = 0) and 1,200 people are infected 7 days later (t = 7), we can plug these values into the equation:
N(0) = N0 * e^(k * 0)
1,000 = N0 * e^(0)
Since any quantity raised to the power of 0 is always equal to 1, we can simplify the equation to:
N(0) = N0 * 1
1,000 = N0
Therefore, the initial number of infected people, N0, is equal to 1,000.
Now, let's use the information that there are 1,200 infected people 7 days later:
N(7) = N0 * e^(k * 7)
1,200 = 1,000 * e^(k * 7)
To isolate the exponential term, we divide both sides of the equation by 1,000:
(1,200 / 1,000) = e^(k * 7)
1.2 = e^(k * 7)
Taking the natural logarithm (ln) of both sides:
ln(1.2) = k * 7
Now, we can solve for the growth rate constant, k:
k = ln(1.2) / 7
k ā 0.02531
Finally, substituting the value of k into the exponential growth model equation, we get:
N(t) = 1,000 * e^(0.02531t)
Therefore, the equation representing the exponential growth model for the epidemic is N(t) = 1,000 * e^(0.02531t).
N(t) = Nā * e^(kt)
Where:
N(t) is the number of people infected at time t (in this case, in days).
Nā is the initial number of infected people when the epidemic is first discovered.
k is the growth rate constant that we need to determine.
e is the mathematical constant approximately equal to 2.71828.
Given that there are 1,000 infected people at the start of the epidemic (t = 0), we have the initial condition:
N(0) = Nā = 1,000
Next, we can use the information that there are 1,200 infected people 7 days later (t = 7):
N(7) = Nā * e^(k * 7) = 1,200
Now, we can solve for k. Dividing both sides of the equation by Nā and taking the natural logarithm (ln) of both sides gives us:
ln(e^(k * 7)) = ln(1,200 / Nā)
Using the property of logarithms, we can simplify:
k * 7 = ln(1,200 / Nā)
To find the value of k, we need to substitute the given value of Nā = 1,000 and calculate:
k * 7 = ln(1,200 / 1,000)
k * 7 = ln(1.2)
Now we can solve for k:
k = ln(1.2) / 7 ā 0.0260459367
So, you were correct in calculating k = 0.0260459367.
Finally, we can write the exponential growth model for the epidemic:
N(t) = 1,000 * e^(0.0260459367 * t)
This equation describes the number of infected people N(t) as a function of time t in days.