To determine how many hours it will take for the number of bacteria to reach 2700 when the population increases according to the function [tex]\( P(h) = 1800 e^{0.18 h} \)[/tex], follow these steps:
1. Understand the given function:
[tex]\( P(h) = 1800 e^{0.18 h} \)[/tex]
where [tex]\( P(h) \)[/tex] represents the population of bacteria at time [tex]\( h \)[/tex] (in hours).
2. Set up the equation:
We need to find [tex]\( h \)[/tex] when [tex]\( P(h) = 2700 \)[/tex]. Substitute 2700 for [tex]\( P(h) \)[/tex]:
[tex]\( 2700 = 1800 e^{0.18 h} \)[/tex]
3. Isolate the exponential term:
Divide both sides of the equation by 1800 to isolate the exponential expression:
[tex]\( \frac{2700}{1800} = e^{0.18 h} \)[/tex]
Simplifying the fraction on the left side:
[tex]\( 1.5 = e^{0.18 h} \)[/tex]
4. Take the natural logarithm of both sides:
Apply the natural logarithm (ln) to both sides to solve for [tex]\( h \)[/tex]:
[tex]\( \ln(1.5) = \ln(e^{0.18 h}) \)[/tex]
Using the property of logarithms [tex]\( \ln(e^x) = x \)[/tex]:
[tex]\( \ln(1.5) = 0.18 h \)[/tex]
5. Solve for [tex]\( h \)[/tex]:
Divide both sides of the equation by 0.18 to isolate [tex]\( h \)[/tex]:
[tex]\( h = \frac{\ln(1.5)}{0.18} \)[/tex]
Calculate [tex]\( \ln(1.5) \)[/tex] first and then perform the division.
Note: Since we are instructed not to round any intermediate computations, consider the precise value obtained here.
6. Result:
The value obtained for [tex]\( h \)[/tex] is approximately [tex]\( 2.252583933934247 \)[/tex].
Rounding this to the nearest tenth:
[tex]\( h \approx 2.3 \)[/tex] hours
Therefore, it will take approximately [tex]\( 2.3 \)[/tex] hours for the number of bacteria to reach 2700.
