Sure, let's determine the equation needed to find out how long the bacteria have been in the dish. Here is a step-by-step explanation:
1. Start with the given data:
- Initial number of bacteria, [tex]\( A_0 = 300 \)[/tex]
- Current number of bacteria, [tex]\( A = 5000 \)[/tex]
- The bacteria triples every [tex]\( T = 3 \)[/tex] hours
2. Write the exponential growth equation:
[tex]\[
A = A_0 \cdot (3^{t/T})
\][/tex]
where [tex]\( t \)[/tex] is the time in hours.
3. Plug in the known values:
[tex]\[
5000 = 300 \cdot (3^{t/3})
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
- Divide both sides by 300:
[tex]\[
\frac{5000}{300} = 3^{t/3}
\][/tex]
- Simplify the left-hand side:
[tex]\[
\frac{5000}{300} = 16.\overline{6}
\][/tex]
So,
[tex]\[
16.\overline{6} = 3^{t/3}
\][/tex]
- To isolate [tex]\( t \)[/tex], take the natural logarithm of both sides:
[tex]\[
\ln(16.\overline{6}) = \ln(3^{t/3})
\][/tex]
- Use the properties of logarithms ([tex]\( \ln(a^b) = b\ln(a) \)[/tex]):
[tex]\[
\ln(16.\overline{6}) = \frac{t}{3} \ln(3)
\][/tex]
- Solve for [tex]\( t \)[/tex]:
[tex]\[
t = 3 \cdot \frac{\ln(16.\overline{6})}{\ln(3)}
\][/tex]
This is the equation that you would use to determine how long the bacteria have been in the dish.
