Question

katie is making her special punch for the starry night homecoming dance. there is a proportional relationship between the number of cans of peach juice concentrate in a punch bowl, x, and the corresponding number of bottles of lemon - lime soda, y.

Explanation:

To solve the problem, we first need to determine the constant of proportionality (slope) for the proportional relationship between \( x \) (cans of peach juice concentrate) and \( y \) (bottles of lemon - lime soda).

Step 1: Identify two points on the line

From the graph, we can see that when \( x = 5\), \( y = 10\) (we can also use the point \((10,y)\) from the graph, but let's use \((5,10)\) for simplicity). For a proportional relationship, the equation is of the form \( y=kx\), where \( k\) is the constant of proportionality (slope).

Step 2: Calculate the constant of proportionality \( k\)

We know that for a proportional relationship \( y = kx\), and we can solve for \( k\) using the formula \( k=\frac{y}{x}\).
Substituting \( x = 5\) and \( y = 10\) into the formula, we get \( k=\frac{10}{5}=2\). So the equation of the line is \( y = 2x\).

If we want to find the number of bottles of lemon - lime soda for a given number of cans of peach juice concentrate or vice - versa, we can use this equation. For example, if we want to find \( y\) when \( x = 10\), we substitute \( x = 10\) into \( y = 2x\), and we get \( y=2\times10 = 20\) (which matches the point on the graph \((10,20)\)).

If the question was to find the constant of proportionality, the answer is \( 2\). If it was to find \( y\) for a given \( x\) or \( x\) for a given \( y\), we can use the equation \( y = 2x\).

Assuming the question is to find the constant of proportionality:

Step 1: Select a point from the graph

We choose the point \((x = 5,y = 10)\) from the graph of the proportional relationship.

Step 2: Calculate the constant of proportionality \(k\)

Using the formula for a proportional relationship \(y=kx\), we solve for \(k\) by rearranging to \(k=\frac{y}{x}\). Substituting \(x = 5\) and \(y = 10\) gives \(k=\frac{10}{5}=2\).

Answer:

The constant of proportionality is \(2\)