Questions: A certain strain of bugs had a population of 72. Two weeks later the population had reduced to 28. (a) Develop the model of the form p = a e^(k t) that represents the population after t weeks.

Solution Steps

To develop the model of the form \( p = a e^{k t} \), we need to determine the constants \( a \) and \( k \). The initial population at \( t = 0 \) is 72, which gives us \( a = 72 \). We also know that after 2 weeks, the population is 28. We can use this information to solve for \( k \) by substituting these values into the equation and solving for \( k \).

The initial population of the bugs at \( t = 0 \) weeks is given as \( p(0) = 72 \). Thus, we have: \[ a = 72 \]

We know that after 2 weeks, the population is \( p(2) = 28 \). We can use the model \( p(t) = a e^{kt} \) to find \( k \). Substituting the known values into the equation: \[ 28 = 72 e^{2k} \] To isolate \( k \), we first divide both sides by 72: \[ \frac{28}{72} = e^{2k} \] This simplifies to: \[ \frac{7}{18} = e^{2k} \] Taking the natural logarithm of both sides gives: \[ \ln\left(\frac{7}{18}\right) = 2k \] Solving for \( k \): \[ k = \frac{1}{2} \ln\left(\frac{7}{18}\right) \approx -0.4722 \]

Now that we have both \( a \) and \( k \), we can express the population model as: \[ p(t) = 72 e^{-0.4722t} \]

Final Answer