Question

problem 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).
vs volume
vs volume
(24,360)
350
300
250
200
150
100
50
(10,150)
(24,4.95)
(10,3.75)
volume (oz)
volume (oz)
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?
problem 2
lin and andre biked home from school at a steady pace. lin biked 1.5 km and it took her 5 minutes. andre biked 2 km and it took him 8 minutes.

Explanation:

Step1: Identify the graphs

The first graph has values in dollars (judging by the small - value range like 3.75 and 4.95), so it is the cost vs volume graph. The second graph has values in the hundreds (150, 360), which are typical calorie values, so it is the calories vs volume graph.

Step2: Check for proportionality

For a proportional relationship, the ratio $\frac{y}{x}$ should be constant. For the calories - volume graph:
For the point $(10,150)$, $\frac{150}{10}=15$. For the point $(24,360)$, $\frac{360}{24} = 15$. The ratio of calories to volume is constant, so calories and volume are in a proportional relationship. The cost - volume graph does not have a constant ratio. For example, for the point $(10,3.75)$, $\frac{3.75}{10}=0.375$ and for $(24,4.95)$, $\frac{4.95}{24}\approx0.206$.

Step3: Find the constant of proportionality

The constant of proportionality $k$ for the calories - volume relationship is found by taking the ratio of calories to volume. Using the point $(10,150)$, $k = 15$. This means that for every 1 ounce increase in the volume of the drink, there are 15 more calories.

Answer:

a. The first graph is "Cost (in dollars) vs volume (in ounces)". The second graph is "Calories vs volume (in ounces)".
b. Calories and volume appear to be in a proportional relationship. We know this because the ratio of calories to volume is constant for different data - points.
c. The constant of proportionality is 15. It means that for each additional ounce of the drink, there are 15 more calories.