Question
Carter is hooked on a new book series, The Galaxy. Each book in the series is the same length and chronicles a different year in the Waka Waka Galaxy. Carter has cleared off the top shelf of his bookcase to leave room for each of the books as they come out.
There is a proportional relationship between the number of books on the shelf, x, and how much shelf space the books take up (in inches), y.
x (books) y (inches)
2 2
11 11
13 13
16 16
What is the constant of proportionality? Write your answer as a whole number or decimal.
inches per book
There is a proportional relationship between the number of books on the shelf, x, and how much shelf space the books take up (in inches), y.
x (books) y (inches)
2 2
11 11
13 13
16 16
What is the constant of proportionality? Write your answer as a whole number or decimal.
inches per book
Answers
GPT-4o mini
To find the constant of proportionality in this relationship, we can use the given pairs of values in the table. The constant of proportionality \( k \) is calculated by dividing the dependent variable \( y \) (inches) by the independent variable \( x \) (number of books):
\[
k = \frac{y}{x}
\]
For the given values:
1. For \( x = 2 \), \( y = 2 \):
\[
k = \frac{2}{2} = 1
\]
2. For \( x = 11 \), \( y = 11 \):
\[
k = \frac{11}{11} = 1
\]
3. For \( x = 13 \), \( y = 13 \):
\[
k = \frac{13}{13} = 1
\]
4. For \( x = 16 \), \( y = 16 \):
\[
k = \frac{16}{16} = 1
\]
In all cases, the constant of proportionality \( k \) is equal to 1. Therefore, the constant of proportionality is:
\[
\boxed{1} \text{ inch per book.}
\]
\[
k = \frac{y}{x}
\]
For the given values:
1. For \( x = 2 \), \( y = 2 \):
\[
k = \frac{2}{2} = 1
\]
2. For \( x = 11 \), \( y = 11 \):
\[
k = \frac{11}{11} = 1
\]
3. For \( x = 13 \), \( y = 13 \):
\[
k = \frac{13}{13} = 1
\]
4. For \( x = 16 \), \( y = 16 \):
\[
k = \frac{16}{16} = 1
\]
In all cases, the constant of proportionality \( k \) is equal to 1. Therefore, the constant of proportionality is:
\[
\boxed{1} \text{ inch per book.}
\]