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

\(450=100e^{0.25h}\)

Step2: Solve for \(e^{0.25h}\)

Divide both sides by \(100\): \(\frac{450}{100}=e^{0.25h}\), so \(4.5 = e^{0.25h}\)

Step3: Take the natural logarithm of both sides

\(\ln(4.5)=\ln(e^{0.25h})\)
Since \(\ln(e^{x})=x\), we have \(\ln(4.5) = 0.25h\)

Step4: Solve for \(h\)

\(h=\frac{\ln(4.5)}{0.25}\)
We know that \(\ln(4.5)\approx1.504\)
Then \(h=\frac{1.504}{0.25}=6.016\approx6\)

Answer:

6 hours