Question

The area covered by a certain population of bacteria increases according to a continuous exponential growth model. Suppose that a sample culture has an initial area of and an observed doubling time of days.

(a)Let be the time (in days) passed, and let be the area of the sample at time . Write a formula relating to . Use exact expressions to fill in the missing parts of the formula. Do not use approximations.
(b)What will the area of the sample be in days

Answer 1

In a population following an exponential growth model, the area it covers can be calculated using the formula A(t) = A(0) × 2^(t/D), where A(t) is the area at time t, A(0) is the initial area, and D is the doubling time.

Step-by-step explanation:

The exponential growth model is used to describe populations that double after a certain constant interval of time. A bacteria population in this context follows the same model, given that it has a doubling time and unlimited resources.

Exponential Growth Formula:

For a population that doubles at regular intervals, the population at any time t can be given by the equation A(t) = A(0) × 2^(t/D), where:

A(t) is the area covered by the bacteria at time t,

A(0) is the initial area covered by the bacteria,

t is the number of days passed, and

D is the doubling time of the bacteria population in days.

Calculation of Area After Certain Days:

If we want to calculate the area covered by the bacteria after a certain number of days, we simply plug in the values into our equation. For t days, the covered area would be the initial area multiplied by 2 raised to the power of t divided by the doubling time D.