Question

The number of bacteria in a sample can be modeled by the equation A = 256e^(1.386t), where t is in hours. In approximately how many hours will there be 65,459 bacteria?

Please calculate and provide an estimate of the time it will take for the sample to have approximately 65,459 bacteria.

Answer 1

To find the approximate time it will take for the sample to have approximately 65,459 bacteria, we can solve the equation A = 256e^(1.386t) = 65,459 for t. The approximate time is 4.42 hours.

Step-by-step explanation:

To find the approximate time it will take for the sample to have approximately 65,459 bacteria, we can set up the equation A = 256e^(1.386t) equal to 65,459 and solve for t.

65,459 = 256e^(1.386t)

Divide both sides by 256:

e^(1.386t) = 65,459 / 256

Take the natural logarithm of both sides:

ln(e^(1.386t)) = ln(65,459 / 256)

Use the property of logarithms to bring the exponent down:

1.386t * ln(e) = ln(65,459 / 256)

ln(e) is equal to 1, so we can simplify:

1.386t = ln(65,459 / 256)

Divide both sides by 1.386 to solve for t:

t = ln(65,459 / 256) / 1.386

Using a calculator, we find that t is approximately 4.42 hours. Therefore, in approximately 4.42 hours, there will be approximately 65,459 bacteria in the sample.