Question

Bacteria Growth The number of bacteria in a petri dish culture after \(t\) hours is $$B=100 e^{0.693 t}$$ (a) What was the initial number of bacteria present? (b) How many bacteria are present after 6 hours? (c) Approximately when will the number of bacteria be 200\(?\) Estimate the doubling time of the bacteria.

Short answer

(a) The initial number of bacteria present was 100. (b) After 6 hours, there are approximately 800 bacteria. (c) The number of bacteria will be 200 approximately after 57 minutes, and the doubling time of the bacteria will be approximately 1 hour.

Step-by-step solution

Identify Initial Number of Bacteria

The initial number of bacteria can be found by setting \(t = 0\) in the equation \(B = 100 e^{0.693 t}\). The reason is that \(t = 0\) represents the initial state or start of the bacteria growth.

Calculate Number of Bacteria After 6 Hours

To calculate the number of bacteria after 6 hours we substitute \(t = 6\) in the equation \(B = 100 e^{0.693 t}\), and compute the value of \(B\).

Determine When Number of Bacteria is 200

To approximate when the number of bacteria will be 200, we substitute \(B = 200\) into the equation \(B = 100 e^{0.693 t}\) and solve for \(t\). This requires an understanding of logarithmic maths, as the equation will need to be converted to a logarithmic form to solve.

Estimate Doubling Time

The doubling time for the bacteria can be calculated by setting \(B = 2P = 2*(100) = 200\) in the exponential growth equation and then finding \(t\). It is the time required for the bacteria population to double.