Worded question need help to solve

Question

After a recycling program was introduced, the weight of rubbish disposed by a organisation each month is given by W = 80(3 ^{- 0.026m} )

W = measured in kg.

m = number of months since the recycling program was introduced.

1. Weight of rubbish disposed before the program started.
2. Weight of rubbish disposed after the 1st year of the program.
3. How long after the recycling starts will the weight of rubbish disposed = 20kg?

Answer 1

1) That's easy enough since m = 0 when program started. W = 80(3^-0.026(0)) = 80(1) = 80, since any no. raised to the 0 power = 1

2) After the 1st yr., m = 12 (months) W = 80(3^-0.026(12)) = 80(3^-0.312) = 80(0.7098) ≅ 56.8 Kg.

3) W = 20 so 20 = 80(3^-0.026m) Can divide both sides by 80 -> 0.25 = 3^-0.026m

Now it becomes a log problem. Take natural log of both sides: ln(0.25) = ln(3^-0.026m)

Can re-write as ln(0.25) = -0.026m(ln3) then divide both sides by ln(3)

ln(0.25)/ln(3) = -0.026m -> -1.3863/1.0986 = -0.026m -> -1.2619 = -0.026m

m ≅ 48.53 mos.

Chk: W = 80(3^{-0.026(48.53)}) -> 80(3^-1.2619) -> 80(0.2499) ≅ 20 Kg. after approx. 48.53 mos.

Answer 2

1) W = 80(3 ^{- 0.026(0)} )

2) W = 80(3 ^{- 0.026(1)} )

3) 20 = 80(3 ^{- 0.026m} )