On the first day of spring, an entire field of flowering trees blossoms. The population of locusts consuming these flowers rapidly increases as the trees blossom. $\newline$ The relationship between the elapsed time, $t$ , in weeks, since the beginning of spring, and the total number of locusts, $N(t)$ , is modeled by the following function: $\newline$ $N(t)=300 \cdot 9^{t}$ $\newline$ Complete the following sentence about the rate of change in the locust population. $\newline$ Round your answer to two decimal places. $\newline$ The number of locusts is tripled every weeks.

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Interpret the exponential function word problem

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Q. On the first day of spring, an entire field of flowering trees blossoms. The population of locusts consuming these flowers rapidly increases as the trees blossom.

\newline

The relationship between the elapsed time,

t

, in weeks, since the beginning of spring, and the total number of locusts,

N(t)

, is modeled by the following function:

\newline

N(t)=300 \cdot 9^{t}

\newline

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

\newline

Round your answer to two decimal places.

\newline

The number of locusts is tripled every weeks.

Set up equation: To find the rate at which the number of locusts is tripled, we need to determine the time it takes for the population to multiply by three according to the given function $N(t) = 300 \times 9^{t}$ .
Solve for t: We set up the equation $300 \times 9^{t} = 3 \times 300$ to find the value of t when the population is tripled. This simplifies to $9^{t} = 3$ .
Calculate t: Taking the logarithm of both sides of the equation, we get $\log(9^{t}) = \log(3)$ . Using the property of logarithms that $\log(a^{b}) = b \cdot \log(a)$ , we can rewrite the equation as $t \cdot \log(9) = \log(3)$ .
Convert to weeks: Now we divide both sides by $\log(9)$ to solve for $t$ : $t = \frac{\log(3)}{\log(9)}$ .
Convert to weeks: Now we divide both sides by $\log(9)$ to solve for $t$ : $t = \frac{\log(3)}{\log(9)}$ .Using a calculator, we find that $\log(3)$ is approximately $0.4771$ and $\log(9)$ is approximately $0.9542$ .
Convert to weeks: Now we divide both sides by $\log(9)$ to solve for $t$ : $t = \frac{\log(3)}{\log(9)}$ .Using a calculator, we find that $\log(3)$ is approximately $0.4771$ and $\log(9)$ is approximately $0.9542$ .We calculate $t = \frac{0.4771}{0.9542}$ , which gives us approximately $t = 0.5$ .
Convert to weeks: Now we divide both sides by $\log(9)$ to solve for $t$ : $t = \frac{\log(3)}{\log(9)}$ . Using a calculator, we find that $\log(3)$ is approximately $0.4771$ and $\log(9)$ is approximately $0.9542$ . We calculate $t = \frac{0.4771}{0.9542}$ , which gives us approximately $t = 0.5$ . Since $t$ is in weeks, the number of locusts is tripled every $t$ $0$ weeks. However, the sentence to complete states ＂The number of locusts is tripled every ___ weeks.＂ Therefore, we need to express the answer as a multiple of a week. Since $t$ $0$ weeks is half a week, the correct completion of the sentence is ＂The number of locusts is tripled every half a week.＂