U.S. waste in tons, %, and years since 1960

[jlaw]

Hello Teachers,

Would someone kindly take the time to check the four answers to this long problem?

Problem: In 1960, the United States generated 87.1 millions tons of municipal solid waste and recovered (Recycled) only
4.3% of it. The amount of waste generated can be modeled by the formula w = 3.14n + 87.1, and the amount recycled can be modeled by the formula w = 0.576n + 3.78, where w = millions of tons and n = numbers of years since 1960.

1. Use the formulas to determine the year in which the United States generated over 100 million tons of municipal solid waste.

Is this right?
100 = 3.14n + 87.1
100 - 87.1 = 3.14n + 87.1 - 87.1
12.9 = 3.14n

12.9 = 3.14 n
3.14 3.14

4 = n
1960 + 4 = 1964 when U.S. reached 100 million tons.

2. a. Find the year in which 13% of the municipal solid waste generated will be reovered.

0.13(3.14n + 87.1) = 0.576n + 3.78
0.41n + 11.3 = 0.576n + 3.78
0.41n + 11.3 - 3.78 = 0.576n
0.41n + 7.52 = 0.576n
7.52 = 0.576n - 0.41n

7.52 = 0.166n
0.166 0.166

45 = n
1960 + 45 = 2005 is when the recycle rate becomes 13%.

2. b) Will the recovery rate ever reach 25%?

0.25(3.14n + 87.1) = 0.576n + 3.78
0.785n + 21.8 = 0.576n + 3.78
0.785n + 21.8 - 3.78 = 0.576n
0.785n + 18.02 = 0.576n
18.02 = 0.576n - 0.785n
18.02 = - 0.209n

18.02 = - 0.209n
0.209 - 0.209

-86 = n
A negative number is not possible when calculating years in this problem. Thus, the recovery rate will never reach 25%.

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

Through trial an error with different percentages (using the same equation methods as above) I got 18% as the maximum recylced waste that will ever be recovered. I also discovered if I divided 0.576n by 3.14n it gives me 18.34%, the exact percentage amount. But I do not know of any other way to find a solution to this question without trial and error. Is there another method using these equations to calculate question number 3 and by chance are any of the above answers right?

Truly Grateful for Your Help,
Julia

 

[Deleted member 4993]

Hello Teachers,

Would someone kindly take the time to check the four answers to this long problem?

Problem: In 1960, the United States generated 87.1 millions tons of municipal solid waste and recovered (Recycled) only
4.3% of it. The amount of waste generated can be modeled by the formula w = 3.14n + 87.1, and the amount recycled can be modeled by the formula w = 0.576n + 3.78, where w = millions of tons and n = numbers of years since 1960.

1. Use the formulas to determine the year in which the United States generated over 100 million tons of municipal solid waste.

Is this right?
100 = 3.14n + 87.1
100 - 87.1 = 3.14n + 87.1 - 87.1
12.9 = 3.14n

12.9 = 3.14 n
3.14 3.14

4 = n
1960 + 4 = 1964 when U.S. reached 100 million tons. <<<< Looks good

2. a. Find the year in which 13% of the municipal solid waste generated will be reovered.

0.13(3.14n + 87.1) = 0.576n + 3.78
0.41n + 11.3 = 0.576n + 3.78
0.41n + 11.3 - 3.78 = 0.576n
0.41n + 7.52 = 0.576n
7.52 = 0.576n - 0.41n

7.52 = 0.166n
0.166 0.166

45 = n
1960 + 45 = 2005 is when the recycle rate becomes 13%. <<<< Looks good

2. b) Will the recovery rate ever reach 25%?

0.25(3.14n + 87.1) = 0.576n + 3.78
0.785n + 21.8 = 0.576n + 3.78
0.785n + 21.8 - 3.78 = 0.576n
0.785n + 18.02 = 0.576n
18.02 = 0.576n - 0.785n
18.02 = - 0.209n

18.02 = - 0.209n
0.209 - 0.209

-86 = n
A negative number is not possible when calculating years in this problem. Thus, the recovery rate will never reach 25%. <<<< Looks good

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

Through trial an error with different percentages (using the same equation methods as above) I got 18% as the maximum recylced waste that will ever be recovered. I also discovered if I divided 0.576n by 3.14n it gives me 18.34%, the exact percentage amount. But I do not know of any other way to find a solution to this question without trial and error. Is there another method using these equations to calculate question number 3 and by chance are any of the above answers right?

Truly Grateful for Your Help,
Julia

\(\displaystyle percent \, \, recovery \, = R \, = \, \frac{(waste recycled)}{(waste generated)} \cdot 100 \, = \, \frac{(0.576n + 3.78)}{(3.14n + 87.1)} \cdot 100\)

\(\displaystyle R \, = \, 18.34 \, - \, \frac{362.02}{3.14n \, + \, 87.1}\)

As you can see from above, the maximum value of 'R' is 18.34 (why?).

 

[jlaw]

\(\displaystyle percent \, \, recovery \, = R \, = \, \frac{(waste recycled)}{(waste generated)} \cdot 100 \, = \, \frac{(0.576n + 3.78)}{(3.14n + 87.1)} \cdot 100\)

\(\displaystyle R \, = \, 18.34 \, - \, \frac{362.02}{3.14n \, + \, 87.1}\)

As you can see from above, the maximum value of 'R' is 18.34 (why?).

Thank You So Very Much Subhotosh Khan for sharing your time and knowledge.

Answering your question above: any percentage higher than 18.34 generates a negative number of years, which is not a feasible solution to the given problem. But looking at the equation above which multiplies 100; 100 is usual considered a whole (or maximum), anything less or more than that does not answer the question. Was that the answer or is it more mathematical? I'm willing to learn how to speak Math language.