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 39.2 barrels, and exactly seven days after the crime, it was estimated at 21.5 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.

Answer 1

1. v(t)=39.2(215/392)^((t-4)/3); v(0) ≈ 87.3 bbl
2. v'(4) ≈ -7.8 bbl/day
3. 18.9 days

Explanation:

You have a chemical that decayed from 39.2 barrels after 4 days to 21.5 barrels after 7 days, and you want to know the initial amount, the rate of change after 4 days, and the number of days until the amount is 2 barrels.

Exponential model

Given two value and times (t₁, v₁) and (t₂, v₂), you can write the exponential function that models this change as ...

`v(t)=v_1\left((v_2)/(v_1)\right)^{(t-t_1)/(t_2-t_1)}\\\\\boxed{v(t)=39.2\left((215)/(392)\right)^{(t-4)/(3)}}`

1) Initial value

The initial amount is v(0):

v(0) = 39.2(215/392)^(-4/3) ≈ 87.3

About 87.3 barrels were initially dumped in the swamp.

2) Rate of change

The rate of change after 4 days is the value of the derivative of the function at that time.

v'(t) = 39.2(1/3)(215/392)^((t-4)/3))·ln(215/392)

v'(4) = 39.2(1/3)·ln(215/392) ≈ -7.8 . . . barrels per day

The Trioxin was decreasing at a rate of 7.8 barrels per day.

3) Time to 2 barrels

We can solve for t:

v(t) = 2 = 39.2(215/392)^((t-4)/3)

2/39.2 = (215/392)^((t-4)/3) . . . . . . . . divide by 39.2

ln(20/392) = (t -4)/3·ln(215/392) . . . . . take logs

t -4 = 3·ln(5/98)/ln(215/392) . . . . . . . . . . divide by the coefficient of t-4

t = 4 +3·ln(5/98)/ln(215/392) ≈ 18.9

It will take about 18.9 days for the amount to drop to 2 barrels.

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Additional comment

Often, an exponential model is of the form ...

v(t) = v₀·e^(kt)

In this form, the model is ...

v(t) = 87.314·e^(-0.20021t) ≈ 87.3e^(-0.2t)