The concentration of a specific chemical pollutant in a reservoir, P P\,P parts per million (ppm), is monitored over time. An environmental scientist models this concentration using the equation
P=12+60(t20)3−15(t20)4 P = 12 + 60\left(\frac{t}{20}\right)^3 - 15\left(\frac{t}{20}\right)^4 P=12+60(20t)3−15(20t)4where t t\,t is the number of years after 1 January 2010.
The scientist predicts the pollutant will be completely dissipated (P=0P=0P=0) at time TTT. Show that T T\,T satisfies the equation
T=80T2+128 000T3 T = \sqrt[3]{80T^2 + \frac{128\,000}{T}} T=380T2+T128000Use the iterative formula Tn+1=80Tn2+128 000Tn3\displaystyle T_{n+1} = \sqrt[3]{80T_n^2 + \frac{128\,000}{T_n}}Tn+1=380Tn2+Tn128000, with T0=75T_0 = 75T0=75, to find the values of T1,T2, T_1, T_2,\,T1,T2, and T3 T_3\,T3 to three decimal places.
Explain the relevance of using T0=75T_0 = 75T0=75 in terms of the date.
On 1 January 2010, a natural neutralizing process in the reservoir begins. Its effectiveness at counteracting the pollutant is modeled by
N=20×1.05t N = 20 \times 1.05^t N=20×1.05tUse the models to show that the pollutant concentration P P\,P and the neutralizing effectiveness N N\,N will be equal during the year 2028.
