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Aleksandra started studying how the number of branches on her tree grows over time. Every 1.5 years, the number of branches increases by an addition of
(2)/(7) of the total number of branches. The number of branches can be modeled by a function,
N, which depends on the amount of time,
t (in years).
When Aleksandra began the study, her tree had 52 branches.
Write a function that models the number of branches
t years since Aleksandra began studying her tree.

N(t)=◻

Aleksandra started studying how the number of branches on her tree grows over time. Every $1$ . $5$ years, the number of branches increases by an addition of $\frac{2}{7}$ of the total number of branches. The number of branches can be modeled by a function, $N$ , which depends on the amount of time, $t$ (in years). $\newline$ When Aleksandra began the study, her tree had $52$ branches. $\newline$ Write a function that models the number of branches $t$ years since Aleksandra began studying her tree. $\newline$ $N(t)=\square$

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Write exponential functions: word problems

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Q. Aleksandra started studying how the number of branches on her tree grows over time. Every

1

.

5

years, the number of branches increases by an addition of

\frac{2}{7}

of the total number of branches. The number of branches can be modeled by a function,

N

, which depends on the amount of time,

t

(in years).

\newline

When Aleksandra began the study, her tree had

52

branches.

\newline

Write a function that models the number of branches

t

years since Aleksandra began studying her tree.

\newline

N(t)=\square

Define Initial Parameters: Let's define the initial number of branches and the growth rate. $\newline$ Initial number of branches $a$ : $52$ $\newline$ Growth rate $r$ : $\frac{2}{7}$
Determine Growth Factor: Determine the growth factor $b$ . Since the number of branches increases by a fraction $\frac{2}{7}$ of the current number, the growth factor is $1$ plus the growth rate. $\newline$ Growth factor $b$ = $1 + r$ $\newline$ $b = 1 + \frac{2}{7}$ $\newline$ $b = \frac{9}{7}$
Write Function for Number of Branches: Now we can write the function that models the number of branches $t$ years since Aleksandra began studying her tree. The function is in the form $N(t) = a(b)^t$ , where $t$ is the time in years. $\newline$ Substitute $52$ for 'a' and $9/7$ for 'b' into the function. $\newline$ $N(t) = 52(9/7)^t$

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