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On day 1 , Jane invites 5 friends to her party. She asks each of those friends to visit and invite 4 of their friends the next day, and to have each of their friends visit and invite 4 of their friends the day after, and so on. Let
P be the number of people who are invited to Jane's party on day
t. Assume that no person is invited more than once. Which of the following best explains the relationship between
t and
P ?
Choose 1 answer:
(A) The relationship is linear because all of the values of
P are multiples of 10 .
(B) The relationship is exponential because
P increases by a factor of 4 each time
t increases by 1 .
(C) The relationship is exponential because
P increases by a factor of 5 each time
t increases by 1 .
D The relationship is linear because
P increases by a factor of 5 as
t changes from 1 to 2 .

On day $1$ , Jane invites $5$ friends to her party. She asks each of those friends to visit and invite $4$ of their friends the next day, and to have each of their friends visit and invite $4$ of their friends the day after, and so on. Let $P$ be the number of people who are invited to Jane's party on day $t$ . Assume that no person is invited more than once. Which of the following best explains the relationship between $t$ and $P$ ? $\newline$ Choose $1$ answer: $\newline$ (A) The relationship is linear because all of the values of $P$ are multiples of $10$ . $\newline$ (B) The relationship is exponential because $P$ increases by a factor of $4$ each time $t$ increases by $1$ . $\newline$ (C) The relationship is exponential because $P$ increases by a factor of $5$ each time $t$ increases by $1$ . $\newline$ (D) The relationship is linear because $P$ increases by a factor of $5$ as $t$ changes from $1$ to $2$ .

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One-step inequalities: word problems

Full solution

Q. On day

1

, Jane invites

5

friends to her party. She asks each of those friends to visit and invite

4

of their friends the next day, and to have each of their friends visit and invite

4

of their friends the day after, and so on. Let

P

be the number of people who are invited to Jane's party on day

t

. Assume that no person is invited more than once. Which of the following best explains the relationship between

t

and

P

?

\newline

Choose

1

answer:

\newline

(A) The relationship is linear because all of the values of

P

are multiples of

10

.

\newline

(B) The relationship is exponential because

P

increases by a factor of

4

each time

t

increases by

1

.

\newline

(C) The relationship is exponential because

P

increases by a factor of

5

each time

t

increases by

1

.

\newline

(D) The relationship is linear because

P

increases by a factor of

5

as

t

changes from

1

to

2

.

Day $1$ Invite: On day $1$ , Jane invites $5$ friends. So, $P = 5$ when $t = 1$ .
Day $2$ Invite: Each friend invites $4$ more friends on the next day. So, on day $2$ , the number of new invites is $5 \times 4$ .
Day $2$ Total: On day $2$ , $P = 5 + (5 \times 4)$ because the original $5$ friends are still counted.
Day $3$ New Invites: On day $3$ , each of the $20$ new people from day $2$ invites $4$ friends, so the number of new invites is $20 \times 4$ .
Day $3$ Total: On day $3$ , $P = 5 + (5 \times 4) + (20 \times 4)$ . This pattern shows that with each day, the number of invites increases by a factor of $4$ times the previous day's new invites.
Relationship Type: The relationship between $t$ and $P$ is not linear because the number of new invites each day is not constant; it's multiplied by $4$ each day.
Exponential Growth: The relationship is exponential because the total number of people invited, $P$ , increases by a factor of $4$ each time $t$ increases by $1$ .

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(C) No, Bridget didn't establish that

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\newline

Below is Ibrahim's attempt to write a formal justification for the fact that the equation

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\newline

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\newline

Ibrahim's justification:

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\newline

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(B) No, Ibrahim didn't establish that

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Below is Amrita's attempt to write a formal justification for the fact that the equation

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\newline

Amrita's justification:

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[-1,4]

g^{\prime}

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(A) Yes, Amrita's justification is complete.

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Below is Omar's attempt to write a formal justification for the fact that there exists a value

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\newline

Is Omar's justification complete? If not, why?

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\newline

f(0)=0

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f

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\newline

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1

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