In $2012,34.1 \%$ of all glass containers were recycled. A beverage company used $450,000 \mathrm{lb}$ of glass containers per year. After recycling, the amount of glass, in pounds, still in use after t years is given by $N(t)=450,000(0.266)^{t}$.
a) Find $N(5)$, and explain its meaning.
b) Find $N^{\prime}(5)$, and explain its meaning.
c) When will $9 \%$ of the original amount of glass still be in use?

Step 1 ：Substitute \(t=5\) into the function \(N(t)=450,000(0.266)^{t}\) to find the amount of glass still in use after 5 years. This gives us \(N(5) = 450,000(0.266)^{5} = 599.27\) pounds.

Step 2 ：Find the derivative of the function \(N(t)\), which represents the rate of change of the amount of glass still in use with respect to time. The derivative is \(N'(t) = -595916.536590197(0.266)^{t}\).

Step 3 ：Substitute \(t=5\) into the derivative function to find the rate of change of the amount of glass still in use after 5 years. This gives us \(N'(5) = -595916.536590197(0.266)^{5} = -793.59\) pounds per year.

Step 4 ：Set the function \(N(t)\) equal to \(9\%\) of the original amount of glass, which is \(450,000 \times 0.09\), and solve for \(t\). This gives us \(t = \log_{0.266}(450,000 \times 0.09 / 450,000) = 1.82\) years.

Step 5 ：Final Answer: a) After 5 years, approximately \(\boxed{599.27}\) pounds of glass are still in use. b) The rate of change of the amount of glass still in use after 5 years is approximately \(\boxed{-793.59}\) pounds per year. c) Approximately \(\boxed{1.82}\) years will pass before \(9\%\) of the original amount of glass is still in use.