What would t = in years

Question

If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula y = y_{0} * e ^ (- 0.3821t) , where the time years and is the pollutant level at which further pollution ceased. How many years would it take to clear 70% of the pollutants? (Round your answer to three decimal places.)

Answer 1

For this kind of problem, we can use the properties of logarithms to simplify the equation and solve for t.

We can plug in 70 for y on the left side of the equation.

70 = y₀e^-0.3821t

Since y₀ is just a constant, let's divide both sides by y₀ so we can have the term involving t isolated.

70/y₀ = e^-0.3821t

Now take the natural log of both sides. (We choose the natural log because it will simplify nicely with e)

ln(70/y₀) = ln(e^-0.3821t)

Next we can use the property of logarithms that allows us to move the exponent to the coefficient.

ln(70/y₀) = -0.3821t*ln(e)

Notice that ln(e) = 1

ln(70/y₀) = -0.3821t

Divide both sides by -03821 so we can isolate t

ln(70/y₀) / -0.3821 = t

The last step would be to plug the left side of this equation into your calculator to find a decimal number equal to t, then round it to 3 decimal places as instructed. Of course, in order to do that, you would need to know what y₀ is equal to.