Lynn uses this equation to model the growth of a bacteria culture. She let N represent the number of bacteria present after t hours of growth.

Question

N equals 100 e Superscript k t Baseline



There were 400 bacteria present after 5 hours of growth.

Question
Which is equal to the value of k?

Answer options with 4 options
A.
Natural Log left-parenthesis StartFraction 4 over 5 EndFraction right-parenthesis EndLog

B.
5 Natural Log 4 EndLog

C.
StartFraction Natural Log 4 EndLog over Natural Log 5 EndLog EndFraction

D.
StartFraction Natural Log 4 EndLog over 5 EndFraction

Answer 1

To find the value of \( k \) in the equation \( N = 100 e^{kt} \) given that there are 400 bacteria present after 5 hours, we can follow these steps:

1. Substitute \( N = 400 \) and \( t = 5 \) into the equation: \[ 400 = 100 e^{5k} \]

2. Divide both sides by 100 to simplify: \[ 4 = e^{5k} \]

3. Take the natural logarithm of both sides: \[ \ln(4) = 5k \]

4. Solve for \( k \): \[ k = \frac{\ln(4)}{5} \]

Thus, the value of \( k \) is: \[ k = \frac{\ln(4)}{5} \]

Among the answer choices provided, the correct option is: D. \( \frac{\ln(4)}{5} \).