Question 1a.
The following equation models the spread of a disease:

Where:
N(t) is the number of people infected at the time.
N0 is the initial number of infected people.
r is the growth rate of the disease.
t is the time in days.
Without substituting the given values for N0 and r immediately, derive a general formula for t in terms of N(t), N0, and r. Using the general formula, calculate the time it would take for 100 people to be infected, assuming N0 =1 and r=0.5.

Question 1b:
Suppose the initial number of infected people (N0) is increased to 3 while maintaining the original growth rate (r=0.5). Calculate the time it would take for 300 people to be infected.

Question 1a: To derive a general formula for t in terms of N(t), N0, and r, we need to rearrange the equation N(t) = N0 * e^(r*t) by taking the natural logarithm of both sides. This gives us:

ln(N(t)) = ln(N0 * e^(r*t))

Using the properties of logarithms, we can simplify this as:

ln(N(t)) = ln(N0) + r*t

Then, we can isolate t by subtracting ln(N0) from both sides and dividing by r. This gives us:

t = (1/r) * (ln(N(t)) - ln(N0))

This is the general formula for t in terms of N(t), N0, and r.

Using the given values of N0 = 1 and r = 0.5, we can plug them into the formula and calculate the time it would take for 100 people to be infected. We get:

t = (1/0.5) * (ln(100) - ln(1))

t = 2 * (4.605 - 0)

t = 9.21

Therefore, it would take 9.21 days for 100 people to be infected, assuming N0 = 1 and r = 0.5.

Question 1b: To calculate the time it would take for 300 people to be infected, assuming N0 = 3 and r = 0.5, we can use the same formula as before, but with different values of N(t) and N0. We get:

t = (1/0.5) * (ln(300) - ln(3))

t = 2 * (5.704 - 1.099)

t = 9.21

Therefore, it would take 9.21 days for 300 people to be infected, assuming N0 = 3 and r = 0.5.