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Bela started studying how the number of branches on her tree grows over time. Every 2.9 years, the number of branches increases by an additional
83%, and can be modeled by a function,
N, which depends on the amount of time,
t (in years).
When Bela began the study, her tree had 60 branches.
Write a function that models the number of branches
t years since Bela began studying her tree.

N(t)=◻

Bela started studying how the number of branches on her tree grows over time. Every $2$ . $9$ years, the number of branches increases by an additional $83 \%$ , and can be modeled by a function, $N$ , which depends on the amount of time, $t$ (in years). $\newline$ When Bela began the study, her tree had $60$ branches. $\newline$ Write a function that models the number of branches $t$ years since Bela began studying her tree. $\newline$ $N(t)=\square$

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Write exponential functions: word problems

Full solution

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

2

.

9

years, the number of branches increases by an additional

83 \%

, and can be modeled by a function,

N

, which depends on the amount of time,

t

(in years).

\newline

When Bela began the study, her tree had

60

branches.

\newline

Write a function that models the number of branches

t

years since Bela began studying her tree.

\newline

N(t)=\square

Identify initial number and growth rate: Identify the initial number of branches and the growth rate. $\newline$ The initial number of branches $a$ is given as $60$ . $\newline$ The growth rate $r$ is $83\%$ , which can be expressed as a decimal by dividing by $100$ : $r = \frac{83}{100} = 0.83$ .
Determine growth factor: Determine the growth factor. $\newline$ Since the number of branches increases by $83\%$ , the growth factor ( $b$ ) is $1$ plus the growth rate: $b = 1 + r$ . $\newline$ $b = 1 + 0.83$ $\newline$ $b = 1.83$
Write exponential function for number of branches: Write the function that models the number of branches. $\newline$ The function $N(t)$ that models the number of branches $t$ years since Bela began studying her tree is in the form of an exponential function: $N(t) = a(b)^t$ . $\newline$ However, since the growth happens every $2.9$ years, we need to adjust the exponent to reflect this. The exponent should be $\frac{t}{2.9}$ to account for the growth period. $\newline$ $N(t) = 60(1.83)^{\frac{t}{2.9}}$

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