Question 4.4.8.6: Wood products The EFISCEN wood product model classifies wood......
Wood products
The EFISCEN wood product model classifies wood products according to their life-span. There are four classifications: short (1 year), medium short (4 years), medium long (16 years), and long (50 years). Based on data obtained from the European Forest Institute, the percentage of remaining wood products after t years for wood products with long life-spans (such as those used in the building industry) is given by
P(t)\,=\,{\frac{100.3952}{1\,+\,0.0316e^{6.0581t}}}(a) What is the decay rate?
(b) What is the percentage of remaining wood products after 10 years?
(c) How long does it take for the percentage of remaining wood products to reach 50%?
(d) Explain why the numerator given in the model is reasonable.
(a) The decay rate is |b|\,=\,|-0.0581| = 5.81%
(b) Evaluate P(10)
P(10)\,=\,{\frac{100.3952}{1\,+\,0.0316e^{0.0581(10)}}}\,\approx\,95.0So 95% of long- life-spanwood products remain after 10 years.
(c) Solve the equation P(t) = 50
{\frac{100.3952}{1+0.0316e^{0.0581t}}}=50 \\ 100.3952=50(1+0.0316e^{0.0581t})2.0079=1+0.0316e^{0.0581t} Divide both sides by 50.
1.0079=0.0316e^{0.0581t} Subtract 1 from both sides.
31.8956=e^{0.0581t} Divide both sides by 0.0316.
\ln{(31.8956)}=0.0581t Rewrite as a logarithmic expression
t ≈ 59.6 years Divide both sides by 0.0581.
It will take approximately 59.6 years for the percentage of long-life-span wood products remaining to reach 50%.
(d) The numerator of 100.3952 is reasonable because the maximum percentage of wood products remaining that is possible is 100%.