Question

a scientist writes the equation ( n(h)=100e^{0.25h} ) to model the growth of a certain bacteria in a petri dish, where ( n ) represents the number of bacteria after ( h ) hours. after approximately how many hours will 450 bacteria be present? round your answer to the nearest whole number. 1 hour 6 hours 13 hours 15 hours

Explanation:

Step1: Substitute \( N(h) = 450 \) into the equation

Given \( N(h)=100e^{0.25h} \), substitute \( N(h) = 450 \). So, \( 450=100e^{0.25h} \).
Divide both sides by 100: \( \frac{450}{100}=e^{0.25h} \), which simplifies to \( 4.5 = e^{0.25h} \).

Step2: Take the natural logarithm of both sides

Using the property \( \ln(e^{x})=x \), if \( 4.5 = e^{0.25h} \), then \( \ln(4.5)=\ln(e^{0.25h}) \).
Since \( \ln(e^{0.25h}) = 0.25h \), we have \( \ln(4.5)=0.25h \).
We know that \( \ln(4.5)\approx1.504 \). So, \( 1.504 = 0.25h \).

Step3: Solve for \( h \)

Divide both sides of the equation \( 1.504 = 0.25h \) by \( 0.25 \).
\( h=\frac{1.504}{0.25}=6.016\approx6 \)

Answer:

6 hours