Question

In 2012, 34.1% of all glass containers were recycled. A beverage company used 200,000 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) = 200,000(0.406) a) Find N(5), and explain its meaning. b) Find N'(5), and explain its meaning. c) When will 4% of the original amount of glass still be in use? a) N(5) = 2206.28 lb (Round to two decimal places as needed.) The expression N(5) means that after 5 year(s), the amount of glass still in use is N(5) pounds. b) N'(5)= - 1988.74 lb (Round to two decimal places as needed.) The expression N'(5) means that after 5 year(s), the amount of glass still in use is changing by N'(5) pounds per year. years. c) Four percent of the original amount of glass will still be in use after (Round to two decimal places as needed.)

Answer 1

a) N(5) represents the amount of glass still in use after 5 years and is equal to 2206.28 lb. b) N'(5) represents the rate at which the amount of glass still in use is changing after 5 years and is equal to -1988.74 lb/year. c) 4% of the original amount of glass will still be in use after approximately 21.37 years.

Step-by-step explanation:

a) To find N(5), we need to substitute t=5 into the equation N(t) = 200,000(0.406). N(5) = 200,000(0.406)^5 = 2206.28 lb. N(5) represents the amount of glass still in use after 5 years.

b) N'(t) represents the rate of change of N(t). To find N'(5), we need to differentiate N(t) with respect to t. N'(t) = 200,000(0.406)(ln(0.406))(0.406)^t. N'(5) = 200,000(0.406)(ln(0.406))(0.406)^5 = -1988.74 lb/year. N'(5) represents the rate at which the amount of glass still in use is changing after 5 years.

c) We want to find the value of t when 4% of the original amount is still in use. This means we need to find t such that N(t) = 0.04 * 200,000. Solving for t, we have 200,000(0.406)^t = 0.04 * 200,000. Simplifying, (0.406)^t = 0.04. Taking the natural logarithm of both sides, t ln(0.406) = ln(0.04). Solving for t, t = ln(0.04) / ln(0.406) = 21.37 years.