Under ideal conditions, Lemna minor (common duckweed) is a fastgrowing fern that can double its area every $2$ days. Assume the growth is unrestricted, and that the duckweed initially covers $10$ square centimeters $\left(\mathrm{cm}^{2}\right)$ in area. Which of the following functions, $F$ , models the area (in $\mathrm{cm}^{2}$ ) the duckweed covers after $d$ days? $\newline$ Choose $1$ answer: $\newline$ (A) $F(d)=10(0.5)^{\frac{d}{2}}$ $\newline$ (B) $F(d)=2(10)^{d}$ $\newline$ (C) $F(d)=10(2)^{d}$ $\newline$ (D) $F(d)=10(2)^{\frac{d}{2}}$

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Q. Under ideal conditions, Lemna minor (common duckweed) is a fastgrowing fern that can double its area every

2

days. Assume the growth is unrestricted, and that the duckweed initially covers

10

square centimeters

\left(\mathrm{cm}^{2}\right)

in area. Which of the following functions,

F

, models the area (in

\mathrm{cm}^{2}

) the duckweed covers after

d

days?

\newline

Choose

1

answer:

\newline

(A)

F(d)=10(0.5)^{\frac{d}{2}}

\newline

(B)

F(d)=2(10)^{d}

\newline

(C)

F(d)=10(2)^{d}

\newline

(D)

F(d)=10(2)^{\frac{d}{2}}

Problem Understanding: Understand the problem. $\newline$ We need to find a function that models the growth of the duckweed, which doubles in area every $2$ days, starting from an initial area of $10\,\text{cm}^2$ .
Analyzing Answer Choices: Analyze the answer choices. $\newline$ (A) $F(d)=10(0.5)^{\frac{d}{2}}$ suggests the area is halved every $2$ days, which is the opposite of what we want. $\newline$ (B) $F(d)=2(10)^d$ suggests the area is multiplied by $10$ raised to the power of the number of days, which is not a doubling every $2$ days. $\newline$ (C) $F(d)=10(2)^d$ suggests the area doubles every day, which is faster than the given rate. $\newline$ (D) $F(d)=10(2)^{\frac{d}{2}}$ suggests the area is multiplied by $2$ raised to the power of half the number of days, which could represent a doubling every $2$ days.
Determining the Correct Model: Determine the correct model by testing the doubling condition. $\newline$ We know that after $2$ days, the area should be double the initial area, which is $20\,\text{cm}^2$ . Let's test option (D) since it seems to fit the doubling condition: $\newline$ $F(2) = 10(2)^{\frac{2}{2}} = 10(2)^1 = 10 \times 2 = 20\,\text{cm}^2$ . $\newline$ This matches the condition that the duckweed doubles its area every $2$ days.
Confirming Other Options: Confirm that the other options do not match the doubling condition. $\newline$ (A) $F(2) = 10(0.5)^{\frac{2}{2}} = 10(0.5)^1 = 10 \times 0.5 = 5 \text{ cm}^2$ , which is incorrect. $\newline$ (B) $F(2) = 2(10)^2$ is clearly much larger than $20 \text{ cm}^2$ , which is incorrect. $\newline$ (C) $F(2) = 10(2)^2 = 10 \times 4 = 40 \text{ cm}^2$ , which is also incorrect.
Concluding the Correct Function: Conclude the correct function. $\newline$ Based on the analysis, the function that correctly models the growth of the duckweed is $D$ $F(d)=10(2)^{\frac{d}{2}}$ .