Question

A culture of bacteria follows the laws of exponential growth. If 470 bacteria are present initially, and there are 720 bacteria after 1 hour, when will the number of bacteria reach 20,000 ? Round to the nearest tenth of an hour. Use the equation: A=A0​e^kt

Answer 1

To determine when the number of bacteria will reach 20,000, we can use the equation for exponential growth. We can find the value of k by substituting the given values into the equation. Then, we can solve for the time when the number of bacteria reaches 20,000 by substituting the known values back into the equation.

Step-by-step explanation:

To determine when the number of bacteria will reach 20,000, we can use the equation for exponential growth: A = A0ekt. We know the initial number of bacteria (A0) is 470 and after 1 hour (t=1), the number of bacteria (A) is 720. Substituting these values into the equation, we can find the value of k. Once we have the value of k, we can use it to solve for t when A=20,000.

Let's find the value of k first. Substitute A0 = 470, A = 720, and t = 1 into the equation:

720 = 470ek

Divide both sides of the equation by 470:

720/470 = ek

Take the natural logarithm (ln) of both sides of the equation:

ln(720/470) = k

Use a calculator to evaluate the right side of the equation, which gives k ≈ 0.406.

Now that we have the value of k, we can use it to solve for t when A = 20,000. Substitute A0 = 470, A = 20,000, and k ≈ 0.406 into the equation:

20,000 = 470e0.406t

Divide both sides of the equation by 470:

20,000/470 ≈ e0.406t

Take the natural logarithm (ln) of both sides of the equation:

ln(20,000/470) ≈ 0.406t

Divide both sides of the equation by 0.406:

t ≈ ln(20,000/470) / 0.406 ≈ 4.973 hours

Therefore, the number of bacteria will reach 20,000 in approximately 4.973 hours.

Learn more about Exponential growth