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Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every 12 weeks, the number of people who receive the email increases by an additional
99%, and can be modeled by a function,
P, which depends on the amount of time,
t (in weeks).
Nicholas initially sent the chain letter to 50 friends.
Write a function that models the number of people who receive the email
t weeks since Nicholas initially sent the chain letter.

P(t)=◻

Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every $12$ weeks, the number of people who receive the email increases by an additional $99 \%$ , and can be modeled by a function, $P$ , which depends on the amount of time, $t$ (in weeks). $\newline$ Nicholas initially sent the chain letter to $50$ friends. $\newline$ Write a function that models the number of people who receive the email $t$ weeks since Nicholas initially sent the chain letter. $\newline$ $P(t)=\square$

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Write exponential functions: word problems

Full solution

Q. Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every

12

weeks, the number of people who receive the email increases by an additional

99 \%

, and can be modeled by a function,

P

, which depends on the amount of time,

t

(in weeks).

\newline

Nicholas initially sent the chain letter to

50

friends.

\newline

Write a function that models the number of people who receive the email

t

weeks since Nicholas initially sent the chain letter.

\newline

P(t)=\square

Identify initial value and growth rate: Identify the initial value $a$ and the growth rate $r$ . $\newline$ The initial number of friends Nicholas sent the chain letter to is $50$ , so $a = 50$ . $\newline$ The growth rate every $12$ weeks is an additional $99\%$ , which means the number of people increases to $199\%$ of the previous amount every $12$ weeks. To express this as a growth factor for the function, we convert the percentage to a decimal. So, $r = 99\% = 0.99$ .
Calculate growth factor: Calculate the growth factor ( $b$ ). $\newline$ The growth factor is $1$ plus the growth rate. Since the growth rate is $0.99$ , we add this to $1$ to get the growth factor. $\newline$ $b = 1 + r$ $\newline$ $b = 1 + 0.99$ $\newline$ $b = 1.99$
Write the function: Write the function using the initial value and the growth factor. $\newline$ The function that models the number of people who receive the email $t$ weeks since Nicholas initially sent the chain letter is in the form $P(t) = a(b)^{(t/k)}$ , where $k$ is the period of the growth cycle, which is $12$ weeks in this case. $\newline$ Substitute $50$ for ' $a$ ', $1.99$ for ' $b$ ', and $12$ for ' $k$ ' into the function. $\newline$ $P(t) = a(b)^{(t/k)}$ $0$

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