Vera is an ecologist who studies the change in the bear population of Siberia over time.

The relationship between the elapsed time, $t$, in years, since Vera began studying the population, and the total number of bears, $N(t)$, is modeled by the following function:
$$
N(t)=2187 \cdot\left(\frac{2}{3}\right)^{t}
$$

Complete the following sentence about the yearly rate of change of the bear population.

Every year, the bear population grows/shrinks $\vee$ by a factor of
$\square$

Question

Vera is an ecologist who studies the change in the bear population of Siberia over time.

The relationship between the elapsed time, $t$, in years, since Vera began studying the population, and the total number of bears, $N(t)$, is modeled by the following function:
$$
N(t)=2187 \cdot\left(\frac{2}{3}\right)^{t}
$$

Complete the following sentence about the yearly rate of change of the bear population.

Every year, the bear population grows/shrinks $\vee$ by a factor of
$\square$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the yearly rate of change of the bear population, we need to calculate the derivative of the function $N(t) = 2187 \cdot \left(\frac{2}{3}\right)^t$. ‖ ‖ step 2 ⋮ The derivative of $N(t)$ with respect to $t$ is $N'(t) = 2187 \cdot \ln\left(\frac{2}{3}\right) \cdot \left(\frac{2}{3}\right)^t$. ‖ ‖ step 3 ⋮ Since $\ln\left(\frac{2}{3}\right)$ is negative, the derivative $N'(t)$ will be negative for all $t$. This means the bear population is decreasing over time. ‖ ‖ step 4 ⋮ The factor by which the population changes each year is the base of the exponential function, $\frac{2}{3}$. ‖ ∻Answer∻ ⚹ Every year, the bear population shrinks by a factor of $\frac{2}{3}$. ⚹ ∻Key Concept∻ ⚹Derivative of Exponential Functions⚹ ∻Explanation∻ ⚹The derivative of an exponential function $a \cdot b^t$ where $a$ and $b$ are constants, and $t$ is the variable, is found by multiplying the function by the natural logarithm of the base $b$. If $b < 1$, the function represents exponential decay, and the population decreases over time.⚹◊What is the formula for the total number of bears, $N(t)$, in Siberia according to the provided mathematical function?⍭ Generate me a similar question◊

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