Question

a dish with 2 grams of nutrient is being used to model the change in a microbe population with limited resources. the growth rate of the population, r, in millions of microbes per hour h hours after introducing the bacteria is given by: r = 42h - 3h². how many hours after introduction does the growth rate become negative?

Explanation:

Step1: Set up the inequality

We want to find when $r<0$. So we set up the inequality $42h - 3h^{2}<0$.

Step2: Factor the inequality

Factor out a common factor of $3h$ from the left - hand side: $3h(14 - h)<0$.

Step3: Find the roots of the corresponding equation

Set $3h(14 - h)=0$. The roots are $h = 0$ and $h=14$ (by setting each factor equal to zero: $3h=0$ gives $h = 0$ and $14 - h=0$ gives $h = 14$).

Step4: Test intervals

We have three intervals to test: $h<0$, $0 < h<14$, and $h>14$.
For $h<0$, if we take $h=-1$, then $3(-1)(14-(-1))=3(-1)(15)= - 45<0$. But time $h$ (in hours) cannot be negative in this context.
For $0 < h<14$, if we take $h = 1$, then $3\times1\times(14 - 1)=3\times1\times13 = 39>0$.
For $h>14$, if we take $h = 15$, then $3\times15\times(14 - 15)=3\times15\times(-1)=-45<0$.

Answer:

14