To solve the given problem, we need to find the value of [tex]\( h \)[/tex] when [tex]\( N(h) = 450 \)[/tex] in the equation [tex]\( N(h) = 100 e^{0.25 h} \)[/tex].
Follow these steps:
1. Set up the equation:
Given that [tex]\( N(h) = 450 \)[/tex],
[tex]\[
450 = 100 e^{0.25h}
\][/tex]
2. Isolate the exponential term:
Divide both sides of the equation by 100 to isolate the exponential term.
[tex]\[
\frac{450}{100} = e^{0.25h} \rightarrow 4.5 = e^{0.25h}
\][/tex]
3. Take the natural logarithm of both sides:
To solve for [tex]\( h \)[/tex], take the natural logarithm (ln) of both sides of the equation.
[tex]\[
\ln(4.5) = \ln(e^{0.25h})
\][/tex]
4. Simplify the logarithmic equation:
Using the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex],
[tex]\[
\ln(4.5) = 0.25h
\][/tex]
5. Solve for [tex]\( h \)[/tex]:
Divide both sides by 0.25 to solve for [tex]\( h \)[/tex].
[tex]\[
h = \frac{\ln(4.5)}{0.25}
\][/tex]
6. Calculate the value of [tex]\( h \)[/tex]:
Using the natural logarithm value of 4.5,
[tex]\[
\ln(4.5) \approx 1.504
\][/tex]
Therefore,
[tex]\[
h \approx \frac{1.504}{0.25} \approx 6.016
\][/tex]
7. Round the result to the nearest whole number:
The nearest whole number to 6.016 is 6.
Therefore, after approximately 6 hours, 450 bacteria will be present.
So, the answer is:
[tex]\[ \boxed{6} \text{ hours} \][/tex]
