Ju Wenjun is an ecologist who studies the change in the tiger population of Siberia over time. $\newline$ The relationship between the elapsed time $t$ , in years, since Ju Wenjun started studying the population, and the number of tigers, $N(t)$ , is modeled by the following function: $\newline$ $N(t)=650 \cdot\left(\frac{16}{25}\right)^{t}$ $\newline$ Complete the following sentence about the rate of change in the tiger population. $\newline$ Round your answer to two decimal places. $\newline$ The number of tigers decays by a factor of $\frac{4}{5}$ every $\square$ years.

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Compound interest: word problems

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Q. Ju Wenjun is an ecologist who studies the change in the tiger population of Siberia over time.

\newline

The relationship between the elapsed time

t

, in years, since Ju Wenjun started studying the population, and the number of tigers,

N(t)

, is modeled by the following function:

\newline

N(t)=650 \cdot\left(\frac{16}{25}\right)^{t}

\newline

Complete the following sentence about the rate of change in the tiger population.

\newline

Round your answer to two decimal places.

\newline

The number of tigers decays by a factor of

\frac{4}{5}

every

\square

years.

Given Function: We are given the function $N(t) = 650 \times \left(\frac{16}{25}\right)^t$ , which models the number of tigers over time. We want to find the time $t$ when the number of tigers decays by a factor of $\frac{4}{5}$ . This means we are looking for the time $t$ when $N(t)$ is equal to $650 \times \frac{4}{5}$ .
Set Up Equation: First, we need to set up the equation to solve for $t$ : $650 \times \left(\frac{16}{25}\right)^t = 650 \times \frac{4}{5}$
Simplify Equation: We can simplify the equation by dividing both sides by $650$ : $\left(\frac{16}{25}\right)^t = \frac{4}{5}$
Use Logarithms: Now we need to solve for $t$ . Since both sides of the equation are powers of $5$ , we can use logarithms to solve for $t$ . We can take the natural logarithm ( $\ln$ ) of both sides: $\newline$ \ln\left(\left(\frac{$16$}{$25$}\right)^t\right) = \ln\left(\frac{$4$}{$5$}\right)
Apply Power Rule: Using the power rule of logarithms, we can bring down the exponent \(t: $t \cdot \ln\left(\frac{16}{25}\right) = \ln\left(\frac{4}{5}\right)$
Solve for t: Now we can solve for $t$ by dividing both sides by $\ln\left(\frac{16}{25}\right)$ : $\newline$ $t = \frac{\ln\left(\frac{4}{5}\right)}{\ln\left(\frac{16}{25}\right)}$
Calculate Value of t: We can now calculate the value of $t$ using a calculator: $\newline$ $t \approx \frac{\ln(0.8)}{\ln(0.64)}$ $\newline$ $t \approx \frac{-0.2231435513}{-0.4462871026}$ $\newline$ $t \approx 0.5$
Round the Answer: We round the answer to two decimal places as instructed: $t \approx 0.50$