Question

Suppose y ( t ) = 40 e 2 t 8 y(t)=40e 2t 8 represents the number of bacteria present at time t t minutes. at what time will the population reach 100 bacteria?

A) t=8 minutes

B) t=16 minutes

C) t=24 minutes

D) t=32 minutes

Answer 1

To find the time at which the population reaches 100 bacteria, we need to solve the equation 100 = 40e^(2t/8) for t. After solving the equation step-by-step, we find that t is approximately 3.02 minutes.

Step-by-step explanation:

To find the time at which the population reaches 100 bacteria, we need to set the equation y(t) = 40e^(2t/8) equal to 100 and solve for t.

Here's how:

1. Set y(t) = 100: 100 = 40e^(2t/8)
2. Divide both sides by 40: 2.5 = e^(2t/8)
3. Take the natural logarithm (ln) of both sides to eliminate the exponential: ln(2.5) = (2t/8) ln(e)
4. Simplify: ln(2.5) = (2t/8)
5. Multiply both sides by 8/2: (8/2) ln(2.5) = t
6. Calculate: t = 4 ln(2.5)

Using a calculator, we find that t ≈ 3.02 minutes. Therefore, the population will reach 100 bacteria at approximately 3.02 minutes.