# Thread: What am I being asked to find in this word problem? - Rate of change.

1. ## What am I being asked to find in this word problem? - Rate of change.

Hey, I have a word problem here, and I am asking to ensure that I am finding the correct thing.

Question: The amount of pollution in a certain lake is $\displaystyle P(t) = (t^{1/4} + 3)^3$, where t is measured in years, and P is measured in parts permillion (ppm). At what rate is the amount of pollution changing after 16 years?

Answer: I am understanding this to be asking me what the instantaneous rate of change is at the 16 year mark. So I find the derivative of $\displaystyle P(t) = (t^{1/4} + 3)^3$ which is $\displaystyle \frac {3(t^{1/4} + 3)^2}{4(t^{3/4})}$. Then I plug t = 16 into this derivative and this is my answer.

Am I doing the what the question asks?

2. Originally Posted by Kakariki
Hey, I have a word problem here, and I am asking to ensure that I am finding the correct thing.

Question: The amount of pollution in a certain lake is $\displaystyle P(t) = (t^{1/4} + 3)^3$, where t is measured in years, and P is measured in parts permillion (ppm). At what rate is the amount of pollution changing after 16 years?

Answer: I am understanding this to be asking me what the instantaneous rate of change is at the 16 year mark. So I find the derivative of $\displaystyle P(t) = (t^{1/4} + 3)^3$ which is $\displaystyle \frac {3(t^{1/4} + 3)^2}{4(t^{3/4})}$. Then I plug t = 16 into this derivative and this is my answer.

Am I doing the what the question asks?
Yes,
the derivative of the amount of pollution gives the rate of change of pollution,
represented by the slope of the tangent to the graph.
placing t=16 into the slope equation (derivative) gives the
rate of change of pollution at the 16 year mark.

3. Thank you for the confirmation that I am indeed doing this question correctly.