Question 424440
This is a problem in exponential decay for which you can use the exponential function: {{{y = ab^x}}}, where the base, b, is between 0 and 1 for exponential decay. x is the number of years given in the problem (x = 8).
Start with the exponential function:
{{{y = ab^x}}} Substitute y = 450, the current number of owls. a = 12,000, the number of owls 8 years ago, and x = 8.
{{{450 = (12000)b^8}}} Solving for b will give you the exponential function for this situation.
{{{b^8 = 450/12000}}} Use your calculator to take the eighth root of both sides.
{{{b = 0.6634}}} Approx. Now you can write the exponential function for this situation:
{{{y = 12000(0.6634)^x}}}
The question you want to answer is - For what value of x (x = number of years) will y (the number of owls remaining) = 1.
{{{1 = 12000(0.6634)^x}}} Solve for x. Divide both sides by 12000.
{{{1/12000 = (0.6634)^x}}} Take the logarithm of both sides.
{{{Log(1/12000) = xLog(0.6634)}}} From the power rule for logarithms.
{{{x = Log(1/12000)/Log(0.6634)}}} Use your calculator to evaluate.
{{{x = 22.8878}}} years.