During a certain epidemic, the number of people that are infected at any time increases at rate proportional to the number of people that are infected at that time. 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later.
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?
or more accurately
i = 1000 e^kt
1200 = 1000 e^7k
e^7k = 1.2
7 k = ln 1.2
k = .0260459367 as you said
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)
To find the exponential growth model for the epidemic, we need an equation of the form:
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 ≈ 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).
To create an exponential growth model, we can start by using the general form:
N(t) = N₀ * e^(kt)
Where:
N(t) is the number of people infected at time t (in this case, in days).
N₀ 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) = N₀ = 1,000
Next, we can use the information that there are 1,200 infected people 7 days later (t = 7):
N(7) = N₀ * e^(k * 7) = 1,200
Now, we can solve for k. Dividing both sides of the equation by N₀ and taking the natural logarithm (ln) of both sides gives us:
ln(e^(k * 7)) = ln(1,200 / N₀)
Using the property of logarithms, we can simplify:
k * 7 = ln(1,200 / N₀)
To find the value of k, we need to substitute the given value of N₀ = 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 ≈ 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.