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)?
What is the range of P(s)?
Answers
GPT-4o mini
The function \( P(s) \) represents the total number of paragraphs Danielle must write after reading \( s \) short stories. Given that she writes a 3-paragraph summary for each story, we can define the function mathematically as follows:
\[
P(s) = 3s
\]
where \( s \) is the number of short stories read.
Since Danielle has been assigned a collection of 18 short stories, the possible values of \( s \) range from 0 (if she hasn't read any stories) to 18 (if she has read all the stories). Therefore, \( s \) can take on any integer value within this range:
\[
s \in [0, 18]
\]
To find the range of \( P(s) \), we need to evaluate \( P(s) \) at the endpoints of this interval:
1. When \( s = 0 \):
\[
P(0) = 3 \times 0 = 0
\]
2. When \( s = 18 \):
\[
P(18) = 3 \times 18 = 54
\]
Since \( s \) can take any integer value from 0 to 18, \( P(s) \) will take every integer value from \( P(0) \) to \( P(18) \). Thus, the function \( P(s) \) will produce all multiples of 3 in the range from 0 to 54.
The range of \( P(s) \) can be represented as:
\[
\text{Range of } P(s) = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54 \}
\]
In interval notation, the range can also be expressed mathematically as:
\[
\text{Range of } P(s) = [0, 54] \text{ in steps of } 3
\]
In conclusion, the range of \( P(s) \) is all multiples of 3 from 0 to 54.
\[
P(s) = 3s
\]
where \( s \) is the number of short stories read.
Since Danielle has been assigned a collection of 18 short stories, the possible values of \( s \) range from 0 (if she hasn't read any stories) to 18 (if she has read all the stories). Therefore, \( s \) can take on any integer value within this range:
\[
s \in [0, 18]
\]
To find the range of \( P(s) \), we need to evaluate \( P(s) \) at the endpoints of this interval:
1. When \( s = 0 \):
\[
P(0) = 3 \times 0 = 0
\]
2. When \( s = 18 \):
\[
P(18) = 3 \times 18 = 54
\]
Since \( s \) can take any integer value from 0 to 18, \( P(s) \) will take every integer value from \( P(0) \) to \( P(18) \). Thus, the function \( P(s) \) will produce all multiples of 3 in the range from 0 to 54.
The range of \( P(s) \) can be represented as:
\[
\text{Range of } P(s) = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54 \}
\]
In interval notation, the range can also be expressed mathematically as:
\[
\text{Range of } P(s) = [0, 54] \text{ in steps of } 3
\]
In conclusion, the range of \( P(s) \) is all multiples of 3 from 0 to 54.