It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 30 - \frac { 7 } { 2 t + 1 }$ thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be $C ( p ) = 0.3 \sqrt { p ^ { 2 } + p + 30 }$ parts per million (ppm) when the population is p thousand. The rate at which the level of pollution is changing with respect to time 3 years from now is about 0.084 ppm per year.

Question

Quizplus · Accepted Answer

To find the rate at which the level of pollution is changing with respect to time, we need to use the chain rule to differentiate $C(p(t))$ with respect to $t$. First, find $p'(t)$ and then $C'(p)$. Evaluate these derivatives at $t = 3$ and $p(3)$, and multiply them to find the rate of change of pollution level with respect to time. The calculation confirms the rate is approximately 0.084 ppm per year.

It Is Estimated That T Years from Now, the Population p(t)=30−72t+1p ( t ) = 30 - \frac { 7 } { 2 t + 1 }p(t)=30−2t+17​

It Is Estimated That T Years from Now, the Population $p ( t ) = 30 - \frac { 7 } { 2 t + 1 }$