Question

Population Dynamics Suppose that $\mathit{d}\mathit{P}\mathbf{/}\mathit{d}\mathit{t}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{15}\mathit{P}\left(t\right)$represents a mathematical model for the growth of a certain cell culture, where $\mathit{P}\left(t\right)$is the size of the culture (measured in millions of cells) at time $\mathit{t}\mathbf{>}\mathbf{0}$(measured in hours). How fast is the culture growing at the time when the size of the culture reaches 2 million cells?

Short answer

Answer:

The growth rate is around 0.3 million per hour.

Step-by-step solution

Define the derivative of a function

The derivative of a function of a real variable in mathematics describes the sensitivity of the function value (output value) to changes in its argument (input value).

Calculus uses derivatives as a fundamental tool. When a derivative of a single-variable function exists at a given input value, it is the slope of the tangent line to the function's graph at that point.

Determine the mathematical model

It is given that $\frac{dP\left(t\right)}{dt}=1.5P\left(t\right)$. When the size of the culture reaches 2 million cells, the value of $P\left(t\right)$becomes 2.

Substitute 2 for $P\left(t\right)$into $\frac{dP\left(t\right)}{dt}=1.5P\left(t\right)$.

role="math" localid="1655705483426" $$\begin{gathered} \frac{dP\left(t\right)}{dt}=0.15\left(2\right) \\ \frac{dP\left(t\right)}{dt}=0.3millionoerhour \end{gathered}$$

Hence, the growth rate is around 0.3 million per hour.