Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

12. In a petri dish, bacteria triples every 3 hours. If there were initially 300 bacteria and there are currently 5000 bacteria, determine the equation that would be used to find how long the bacteria have been in the dish.

Given:
[tex]\[ A_0 = 300 \][/tex]
[tex]\[ A = 5000 \][/tex]
[tex]\[ T = 3 \text{ hours} \][/tex]

Using the formula for exponential growth:
[tex]\[ A = A_0 \cdot (3)^{\frac{t}{T}} \][/tex]

Where:
[tex]\[ A = \text{final amount of bacteria} \][/tex]
[tex]\[ A_0 = \text{initial amount of bacteria} \][/tex]
[tex]\[ T = \text{time period in which the bacteria triples} \][/tex]
[tex]\[ t = \text{total time elapsed} \][/tex]

Plugging in the given values:
[tex]\[ 5000 = 300 \cdot (3)^{\frac{t}{3}} \][/tex]

Solve for [tex]\( t \)[/tex].


Sagot :

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.