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 = y0e^(−0.3821t),
where t is the time in years and y0 is the pollutant level at which further pollution ceased. How many years would it take to clear 45% of the pollutants? (Round your answer to three decimal places.)
this is just like the interest problem. If 45% are cleared, then 55% are left.
e^(−0.3821t) = 0.55
Now just solve for t
Well, clearing pollutants from Lake Erie sounds like a serious task, but don't worry, I've got the answer for you!
According to the formula y = y0e^(-0.3821t), we're looking for the time when 45% of the pollutants are cleared. In other words, we need to find the value of t when y = 0.45y0.
Let's plug in the values and solve for t:
0.45y0 = y0e^(-0.3821t)
Dividing both sides of the equation by y0:
0.45 = e^(-0.3821t)
To isolate t, we take the natural logarithm (ln) of both sides:
ln(0.45) = -0.3821t
Now, we can solve for t:
t = ln(0.45) / -0.3821
Using a calculator, we find that t is approximately equal to 1.393 years.
So, it would take approximately 1.393 years to clear 45% of the pollutants from Lake Erie.
To find the number of years it would take to clear 45% of the pollutants, we need to solve for t in the equation y = y0e^(-0.3821t) where y is 45% (or 0.45) of the initial pollutant level y0.
Let's substitute these values into the equation:
0.45y0 = y0e^(-0.3821t)
Now, we can simplify the equation by dividing both sides by y0:
0.45 = e^(-0.3821t)
To isolate the exponential term, we take the natural logarithm (ln) of both sides:
ln(0.45) = ln(e^(-0.3821t))
Since ln(e^x) = x, this becomes:
ln(0.45) = -0.3821t
Next, divide both sides by -0.3821:
t = ln(0.45) / -0.3821
Using a calculator, we can evaluate ln(0.45) / -0.3821 ≈ 2.663.
Therefore, it would take approximately 2.663 years to clear 45% of the pollutants.
To find the number of years it would take to clear 45% of the pollutants in Lake Erie, we need to solve for t in the equation:
y = y0e^(-0.3821t)
Given that y is the pollutant level at which further pollution ceased (y0), we know that when y = 0.45y0, it represents 45% of the pollutants being cleared.
Therefore, we can substitute this value into the equation:
0.45y0 = y0e^(-0.3821t)
Next, we can cancel out y0 from both sides of the equation:
0.45 = e^(-0.3821t)
Now, take the natural logarithm (ln) of both sides of the equation to isolate t:
ln(0.45) = ln(e^(-0.3821t))
Using the property of logarithms that ln(e^x) = x, we can simplify the equation further:
ln(0.45) = -0.3821t
Now we can solve for t by dividing both sides of the equation by -0.3821:
t = ln(0.45) / -0.3821
Using a calculator to evaluate ln(0.45) / -0.3821, we find:
t ≈ 2.329 years
Rounding to three decimal places, it would take approximately 2.329 years to clear 45% of the pollutants in Lake Erie if the pollution were stopped suddenly.