Question

name: aaronperiod: 7date: 2/12/24lesson 13 cooldownthe growth of a bacteria population is modeled by the equation $p(h)=1,000e^{0.4h}$. for each question, explain or show how you know.a. how long does it take for the population to double?b. what does the 0.4 mean in context?

Explanation:

Step1: Set up doubling equation

We start with the population model \(p(h) = 1000e^{0.4h}\). To find when the population doubles, set \(p(h) = 2000\):
$$2000 = 1000e^{0.4h}$$

Step2: Simplify the equation

Divide both sides by 1000:
$$2 = e^{0.4h}$$

Step3: Convert to logarithmic form

Take the natural log of both sides:
$$\ln(2) = 0.4h$$

Step4: Solve for h

Isolate \(h\) by dividing both sides by 0.4:
$$h = \frac{\ln(2)}{0.4}$$
Calculate the value: \(h \approx \frac{0.6931}{0.4} \approx 1.73\)

Answer:

a. Approximately 1.73 hours
b. The 0.4 is the continuous growth rate of the bacteria population, meaning the population grows at an instantaneous rate of 40% per hour.