The annual amount of aluminum recycled in a small town is given by the function T(p) = √(1.5p² + 0.25)tons, where p is the population in 1000s. It is estimated that tyears from now, the population of the town will be p(t) = 0.2(t – 1)² – 3 thousand.

How many tons of aluminum are expected to be recycled 6 years from now? Answer with the letter A, B, C, D, or E, not the number, in the blank space provided.
A) 2.5 tons
B) 2 tons
C) 7.37 tons
D) 6.14 tons
E) 0.72 tons
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The yearly amount of aluminum that gets recycled changes. Find the rate of change of the amount with respect to time. Answer with the letter A, B, C, D, or E, not the number, in the blank space provided.
A) 15 tons/yr
B) 0.4 tons/yr
C) 1.2 tons/yr
D) 2.4 tons/yr
E) 0.72 tons/yr
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Given the number of tons of recycled aluminum as a function of population is T(p) = √(1.5p² +0.25), and the estimated population is p(t) = 0.2(t -1)² -3, you want the tons of aluminum recycled 6 years from now, and its rate of change.

Quantity

The population in 6 years is estimated as ...

p(6) = 0.2(6 -1)² -3 = 2 . . . . . thousands

The amount of recycling that population does is ...

T(2) = √(1.5(2²) +0.25) = √6.25 = 2.5 . . . . tons

Six years from now, 2.5 tons of aluminum are expected to be recycled.

Rate of change

The rate of change of the quantity recycled is given by the derivative of the function:

T'(p) = 1.5(2p)/(2√(1.5p² +0.25))·p' = 1.5p·p'/T(p)

In 6 years, this is ...

T'(6) = 1.5·2·p'/2.5 = 1.2p'

The rate of change of population in 6 years is expected to be ...

p'(t) = 0.2(2)(t -1) = 0.4(t -1)

p'(6) = 0.4(6 -2) = 2

So, the rate of change of recycled amount is ...

T'(6) = 1.2(2) = 2.4 . . . . . tons/year

Six years from now the rate of change of recycled amount will be about 2.4 tons per year.

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Additional comment

The rate of change of recycled amount varies substantially. The problem statement does not give a time frame for the desired rate of change. We have assumed it is 6 years, since the value at that time matches one of the answer choices.

The derivative of T has units of tons per thousand population. Since we want tons per year, we need to include the effect of population on the derivative.