At what rate is the amount of material growing if bacteria in a sample increases to 250 when the sample is observed for 150 minutes?

To find the rate at which the amount of material is growing, we need to calculate the derivative of the amount of material with respect to time. In this case, the amount of material is represented by the number of bacteria, and time is represented in minutes.

To begin, let's define some variables:
- Let N(t) represent the number of bacteria at time t.
- Let t represent time in minutes.

We're given that when t = 150, N(t) = 250. We want to find the rate at which the amount of material is growing, which can be represented as dN/dt or N'(t).

To find the derivative, we need an equation that represents the growth of the bacteria over time. The most common model for bacterial growth is the exponential growth model, which states that the rate of change of the number of bacteria is proportional to the number of bacteria itself.

A general form of this model is:
dN/dt = k * N

Where k is a constant that represents the growth rate or the rate at which the bacteria population increases.

Now, we can use the given information to find the value of k. Substituting the given values into the exponential growth model:

dN/dt = k * N
dN/dt = k * 250

We know that when t = 150, N(t) = 250. Therefore, we can substitute these values into the equation:

k * 250 = N'(150)

Solving for k:

k = N'(150) / 250

Finally, to find the rate at which the amount of material is growing when N = 250, we substitute the value of k into the exponential growth model:

dN/dt = k * N
dN/dt = (N'(150) / 250) * N

Now you can substitute the values of N and t into the equation to find the rate at which the amount of material is growing.