A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 290. (a) Find an expression for the number of bacteria after t hours. P(t) = (b) Find the number of bacteria after 2 hours. (Round your answer to the nearest whole number.) P(2) = bacteria (c) Find the rate of growth after 2 hours. (Round your answer to the nearest whole number.) P'(2) = bacteria per hour (d) When will the population reach 10,000? (Round your answer to one decimal place.) t = hr

a)

b) P(2) = 841 bacteria

c) P'(2) = 895 bacteria per hour

d) t = 4.3hr

Step-by-step explanation:

The equation of growth for the number of bacteria can be modeled by the following differential equation:

With solution.

In which P(t) is the population after t hours, P(0) is the initial population and r is the growth rate.

(a) Find an expression for the number of bacteria after t hours.

Initially 100 cells, so P(0) = 100.

After 1 hour, 290 cells, so P(1) = 290.

We apply this to the equation to find the value of r.

Applying ln to both sides of the equality.

So

(b) Find the number of bacteria after 2 hours.

This is P(2).

841 bacteria after 2 hours.

(c) Find the rate of growth after 2 hours.

(d) When will the population reach 10,000?

This is t when P(t) = 10,000. So

Applying ln to both sides

5/12, or 5: 12

step-by-step explanation: