Ju Wenjun is an ecologist who studies the change in the tiger population of Siberia over time. 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: N(t)=650*((16)/(25))^(t) Complete the following sentence about the rate of change in the tiger population. Round your answer to two decimal places. The number of tigers decays by a factor of (4)/(5) every ◻ years.

Question

Ju Wenjun is an ecologist who studies the change in the tiger population of Siberia over time. 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:  N(t)=650*((16)/(25))^(t) Complete the following sentence about the rate of change in the tiger population. Round your answer to two decimal places. The number of tigers decays by a factor of  (4)/(5) every  ◻ years.

Bytelearn · Accepted Answer

Given Function: We are given the function N(t) = 650 * ((16)/(25))^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 (4)/(5). This means we are looking for the time t when N(t) is equal to 650 * (4)/(5).Set Up Equation: First, we need to set up the equation to solve for t: 650 * ((16)/(25))^t = 650 * (4)/(5)Simplify Equation: We can simplify the equation by dividing both sides by 650: ((16)/(25))^t = (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: ln(((16)/(25))^t) = ln((4)/(5))Apply Power Rule: Using the power rule of logarithms, we can bring down the exponent t: t * ln((16)/(25)) = ln((4)/(5))Solve for t: Now we can solve for t by dividing both sides by ln((16)/(25)): t = ln((4)/(5)) / ln((16)/(25))Calculate Value of t: We can now calculate the value of t using a calculator: t ≈ ln(0.8) / ln(0.64) t ≈ -0.2231435513 / -0.4462871026 t ≈ 0.5Round the Answer: We round the answer to two decimal places as instructed: t ≈ 0.50

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