Math  /  Algebra

QuestionDue primarily to irresponsible use of chemicals, otter population in a country declined dramatically during the twentieth century until otters could be found on only a small percentage of riverbanks. After chemical bans were put in place, the number of riverside sites R(t)R(t) occupied by otters rose significantly and can be approximated by the exponential function, R(t)=277(1.071)tR(t)=277(1.071)^{t}, where tt is the number of years since 1985. a) How many riverbanks were occupied by otters in 2010? b) In what year were there otters on 1000 riverbanks? c) What is the doubling time for the number of riverbanks occupied by otters? a) There were \square riverbanks occupied by otters in the year 2010. (Round to the nearest whole number as needed.)

Studdy Solution
Calculate the doubling time for the number of riverbanks occupied by otters.
The doubling time T T can be found using the formula:
(1.071)T=2 (1.071)^T = 2
Take the natural logarithm of both sides:
ln(1.071T)=ln(2) \ln(1.071^T) = \ln(2)
Tln(1.071)=ln(2) T \ln(1.071) = \ln(2)
Solve for T T :
T=ln(2)ln(1.071) T = \frac{\ln(2)}{\ln(1.071)}
Calculate T T :
T0.6930.068610.1 T \approx \frac{0.693}{0.0686} \approx 10.1
Therefore, the doubling time is approximately 10 \boxed{10} years.

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