Question
Danielle has just been assigned a new book to read for her literature class. The book is a collection of 18 short stories. Danielle must also write a 3-paragraph summary after reading each of the short stories. The function P(s) represents the total number of paragraphs Danielle must write, after having read s short stories. What is the range of P(s)?
Answers
GPT-4o mini
To determine the range of the function \( P(s) \), we first need to understand how \( P(s) \) is constructed based on the information provided.
Danielle must write a 3-paragraph summary after reading each short story. Therefore, if she reads \( s \) short stories, the total number of paragraphs she must write is given by the formula:
\[
P(s) = 3s
\]
Now, given that the collection consists of 18 short stories, \( s \) can take on values from 0 (if she has read no stories yet) to 18 (after she has read all the stories). Thus, \( s \) can be expressed as:
\[
s \in \{0, 1, 2, \ldots, 18\}
\]
Accordingly, we can calculate \( P(s) \) for the minimum and maximum values of \( s \):
- When \( s = 0 \):
\[
P(0) = 3 \times 0 = 0
\]
- When \( s = 18 \):
\[
P(18) = 3 \times 18 = 54
\]
Since \( s \) can take any integer value from 0 to 18, \( P(s) \) will take on values from \( 0 \) up to \( 54 \) in steps of 3 (as \( P(s) \) increments by 3 for each additional story read).
Thus, the possible values of \( P(s) \) are:
\[
\{0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54\}
\]
This shows that the total number of paragraphs Danielle must write can range from 0 to 54.
Therefore, the range of \( P(s) \) is:
\[
\text{Range of } P(s) = [0, 54]
\]
This represents all integer multiples of 3 from 0 to 54, inclusive.
Danielle must write a 3-paragraph summary after reading each short story. Therefore, if she reads \( s \) short stories, the total number of paragraphs she must write is given by the formula:
\[
P(s) = 3s
\]
Now, given that the collection consists of 18 short stories, \( s \) can take on values from 0 (if she has read no stories yet) to 18 (after she has read all the stories). Thus, \( s \) can be expressed as:
\[
s \in \{0, 1, 2, \ldots, 18\}
\]
Accordingly, we can calculate \( P(s) \) for the minimum and maximum values of \( s \):
- When \( s = 0 \):
\[
P(0) = 3 \times 0 = 0
\]
- When \( s = 18 \):
\[
P(18) = 3 \times 18 = 54
\]
Since \( s \) can take any integer value from 0 to 18, \( P(s) \) will take on values from \( 0 \) up to \( 54 \) in steps of 3 (as \( P(s) \) increments by 3 for each additional story read).
Thus, the possible values of \( P(s) \) are:
\[
\{0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54\}
\]
This shows that the total number of paragraphs Danielle must write can range from 0 to 54.
Therefore, the range of \( P(s) \) is:
\[
\text{Range of } P(s) = [0, 54]
\]
This represents all integer multiples of 3 from 0 to 54, inclusive.