Question

1. the graphs below show some data from a coffee shop menu. one of the graphs shows cost (in dollars) vs. drink volume (in ounces), and one of the graphs shows calories vs. drink volume (in ounces).

a. which graph is which? give them the correct titles.

b. which quantities appear to be in a proportional relationship? explain how you know.

c. for the proportional relationship, find the constant of proportionality. what does that number mean?

Explanation:

Step1: Identify the graphs

• The left - hand graph has values like \(3.75\) and \(4.95\) which are in the range of dollar amounts. So, the left - hand graph is “cost vs volume”.
• The right - hand graph has values like \(150\) and \(360\) which are in the range of calorie amounts. So, the right - hand graph is “calories vs volume”.

Step2: Check for proportionality

• For a proportional relationship \(y = kx\), where \(k=\frac{y}{x}\).
• For the “calories vs volume” graph:
• When \(x = 10\), \(y=150\), then \(k_1=\frac{150}{10}=15\).
• When \(x = 24\), \(y = 360\), then \(k_2=\frac{360}{24}=15\).
• Since \(\frac{y}{x}\) is constant (\(k = 15\)), calories and volume are in a proportional relationship.

Step3: Find the constant of proportionality

• The constant of proportionality \(k\) for the proportional relationship (calories and volume) is found by \(k=\frac{y}{x}\). Using the point \((10,150)\), \(k=\frac{150}{10}=15\).
• The meaning of \(k = 15\) is that there are 15 calories per ounce of the drink.

Answer:

a. The left - hand graph is “cost vs volume” and the right - hand graph is “calories vs volume”.
b. Calories and volume appear to be in a proportional relationship. Because for the points \((10,150)\) and \((24,360)\) in the “calories vs volume” graph, \(\frac{150}{10}=\frac{360}{24} = 15\) (the ratio of \(y\) (calories) to \(x\) (volume) is constant).
c. The constant of proportionality is \(15\). It means there are 15 calories per ounce of the drink.