Using model, find max. percentage that will be recovered

[lcortina]

I have never used this website before so I hope I understand how this works. I am having difficulties translating or solving this problem.

Problem: In 1960, the United States generated 87.1 million tons of municipal solid waste and recovered (recycled) only 4.3% of it. The amount of municipal solid waste generated in the United States can be modeled by the formula , while the amount recovered can be modeled by the formula , where w is in millions of tons and n is the number of years since 1960.

According to this model, what is the maximum percentage of solid waste that will ever be recovered?

 

[chivox]

Re: Equations and Inequalities

The formulas in this case, if Web sites are to be trusted, are:

(million tons of solid waste generated) = 3.14 n + 87.1

(million tons recovered) = 0.576 n + 3.78

where n is the number of years since 1960 (i.e., 1964's n value for substitution into the equations would be 4). For example, in 1964, the US generated

3.14 (4) + 87.1 = 99.7 million tons of solid waste. That is a lot of garbage.

The good environmentalists recovered 0.576 (4) + 3.78 = 6.08 million tons, which is a pitiful example of environmental clean-up, in my opinion.

OK. Enough with the jokes (sorry). This question is a limit problem. You are asked what is the maximum percentage of solid waste the US will ever recover. What is the mathematical representation of the percentage the US will recover in any given year? You know how many tons it recovers and you know the total amount of solid waste generated. The percentage is just the ratio of those two numbers.

(0.576 n + 3.78) / (3.14 n + 87.1)

would be the percentage of solid waste recovered by the US in whatever year n is. Find the maximum value that this expression could ever have, and there's your answer. You will find it in about 400 years, so I would not recommend the table method. You're going to have to do some algebra. In 2360, for example, the US will generate by this formula 1.34 billion tons of solid waste and recover 234.2 million tons of it. (We will probably have figured out a better way by that time, one would hope.) Anyway, that is about 17.4 percent, which is very close to the limit of the above expression as n approaches infinity.

When finding the limit like this, the constant term is pretty much insignificant, and 0.576 / 3.14 (the coefficients of the independent variable) is about 18.34 percent.

-Paul

 

[stapel]

The amount of municipal solid waste generated in the United States can be modeled by the formula , while the amount recovered can be modeled by the formula , where w is in millions of tons and n is the number of years since 1960.

When you did your copy-n-paste from your online assessment, you neglected to proof-read and include the formulas (which were likely provided as images). Did the tutor guess the right formulas, or has this online exercise been updated since the paid solutions were provided for those particular formulas?

Eliz.

 

[Denis]

Re: Equations and Inequalities

The formulas in this case, if Web sites are to be trusted, are:
(million tons of solid waste generated) = 3.14 n + 87.1

the "3.14" keeps going in circles ? :idea: