Authorities conducted an investigation and found that the chemical Trioxin was i
ID: 2884225 • Letter: A
Question
Authorities conducted an investigation and found that the chemical Trioxin was illegally dumped into the local swamp. Exactly four days after the assumed time of the crime, the amount of Trioxin in the swamp was estimated at about 39.9 barrels, and exactly seven days after the crime, it was estimated at 22 barrels. Assume that no Trioxin was in the swamp before the crime, and that Trioxin decays exponentially. Construct a model that estimates the amount of Trioxin(in barrels)remaining in the swamp t days after it was dumped, then use that model to answer the following. 1) According to the model, how much Trioxin was initially dumped into the swamp? ROUND TO ONE DECIMAL PLACE. About barrels were initially dumped into the swamp. 2) According to the model, at what rate was the amount of Trioxin in the swamp decreasing exactly 4 days after it was dumped? ROUND TO ONE DECIMAL PLACE. The Trioxin was decreasing at a rate of barrels per day. 3) How many days after the crime will/did it take for the amount of Trioxin in the swamp to drop to 2 barrels? ROUND TO ONE DECIMAL PLACE. It took or will take about | days.Explanation / Answer
Since the trioxin was decaying exponentially, we can say its rate of decay as e-kt
Let the initial amount of Trioxin be T0;
Tn ( trioxin on n'th day) = T0 e-kt
Tn= T0 e-kt ( 't' is the day and 'k' is the rate of decay)
39.9 = T4 = To e-k4
22 = T7 = To e-k7
Dividing these two equations we get:
39.9/22 = e-4k/ e-7k
e-k(4-7) = 1.8136
e3k = 1.8136
3k = ln 1.8136
k = 0.595/3 = 0.1984;
Thus k = 0.1984;
T4 = T0 e-0.1984* 4 = 39.9
T0 = 39.9 e0.794
T0 = 39.9 * 2.221 = 88.621;
Thus, the number of barrels initially dumped in the swamp = 88.621 barrels
After 4 days, the rate of decrease was 'k' barrels = 0.1984 barrels a day;
Tn = 2 barrels = T0 e-kt
2= 88.621 e-0.1984 t
e0.1984t = 44.311
0.1984 t = 3.791
t= 3.791/ 0.1984 = 19.109
Thus, after 19.109 days there will be 2 barrels of trioxin left
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