To find the constant of proportionality [tex]\( r \)[/tex] in the equation [tex]\( y = rx \)[/tex] given that the quantities [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are proportional, we need to follow these steps:
1. Identify the given pairs of values:
[tex]\[
\begin{array}{cc}
x & y \\
\hline
5.8 & 5.8 \\
7.5 & 7.5 \\
11.2 & 11.2 \\
\end{array}
\][/tex]
2. Write the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Since [tex]\( y \)[/tex] is proportional to [tex]\( x \)[/tex], we can express this relationship as:
[tex]\[
y = rx
\][/tex]
where [tex]\( r \)[/tex] is the constant of proportionality.
3. Calculate the constant of proportionality [tex]\( r \)[/tex] using any pair of values:
Let's use the first pair [tex]\((5.8, 5.8)\)[/tex]:
[tex]\[
y = rx \implies 5.8 = r \cdot 5.8
\][/tex]
Solve for [tex]\( r \)[/tex] by dividing both sides by 5.8:
[tex]\[
r = \frac{5.8}{5.8}
\][/tex]
4. Simplify the expression:
[tex]\[
r = 1.0
\][/tex]
Thus, the constant of proportionality [tex]\( r \)[/tex] is:
[tex]\[
r = 1.0
\][/tex]
