Question

You prescribe an antibiotic to a patient to cure an infection. We know that the rate of change of bacteria population is proportional to the amount left. Let p(t) represent the population of the bacteria. (a) Write a differential equation for p(t) using k as your constant of proportionality. (b) Solve the differential equation given that originally there were approximately 190 bacteria in the petri dish, but 14 hours later, there were approximately 280 bacteria. (c) How many bacteria can we expect to have after 40 hours? Round to the nearest whole number. (d) When are there approximately 1200 bacteria? Round to three decimal places.

Answer 1

The differential equation for the bacteria population is dp/dt = -kp(t). The population after 14 hours is used to find the value of k. Using the initial population and the value of k, we can find the population after 40 hours. The time when there are approximately 1200 bacteria can also be found.

Step-by-step explanation:

(a) To write a differential equation for the population of bacteria, we know that the rate of change of the bacteria population, dp/dt, is proportional to the amount of bacteria left, p(t). We can write this as: dp/dt = -kp(t), where k is the constant of proportionality.

(b) To solve the differential equation, we need to find the value of k. Given that originally there were approximately 190 bacteria and 14 hours later there were approximately 280 bacteria, we can use the equation p(t) = Pe^(-kt) to find k. Plugging in the values, we get 190 = 280e^(-14k). Solve for k to get k ≈ 0.057.

(c) To find the number of bacteria after 40 hours, we can use the equation p(t) = Pe^(-kt) with the initial population P and the value of k we found. Plugging in the values, we get p(40) ≈ 190e^(-0.057(40)). Round the result to the nearest whole number.

(d) To find when there are approximately 1200 bacteria, we can use the equation p(t) = Pe^(-kt) and solve for t. Plugging in the values, we get 1200 = Pe^(-kt). Solve for t to get the time when there are approximately 1200 bacteria.