Question

homework assignment 6.7 exponential and logarithmic model
due friday by 11:59pm points 10 submitting an external tool
homework assignment 6.7 exponential and logarithmic models
score: 4.67/10 answered: 5/10
question 6
a bacteria culture initially contains 1500 bacteria and doubles every half hour.
find the size of the baterial population after 40 minutes.
find the size of the baterial population after 4 hours.
question help: video
read
written example
submit question

Explanation:

Step1: Determine the exponential growth formula

The general formula for exponential growth is \(P(t)=P_0\cdot a^{\frac{t}{T}}\), where \(P_0\) is the initial population, \(a\) is the growth factor, \(t\) is the time elapsed, and \(T\) is the time it takes for the quantity to change by a factor of \(a\). Here, \(P_0 = 1500\), \(a = 2\) (since the population doubles), and \(T=30\) minutes.

Step2: Calculate the population after 40 minutes

Substitute \(t = 40\) into the formula \(P(t)=1500\cdot2^{\frac{t}{30}}\).

$$
LATEXBLOCK0
$$

Step3: Calculate the population after 4 hours

First, convert 4 hours to minutes. Since \(1\) hour \( = 60\) minutes, \(t=4\times60 = 240\) minutes.
Substitute \(t = 240\) into the formula \(P(t)=1500\cdot2^{\frac{t}{30}}\).

$$
LATEXBLOCK1
$$

Answer:

The size of the bacterial population after 40 minutes is approximately \(3780\). The size of the bacterial population after 4 hours is \(384000\).