Question

carlos runs a lemonade stand. he sells lemonade in different cup sizes and posts this price list:
carlos creates a graph showing the price for each cup size.

1. is the price the same per ounce for any size of cup?
2. what is the slope (unit rate) of the graph?
3. what does the slope mean in this context?

Explanation:

Step1: Calculate the unit rate (slope)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points \((x_1,y_1)=(8,2)\) and \((x_2,y_2)=(12,3)\).

$$m=\frac{3 - 2}{12 - 8}=\frac{1}{4}=0.25$$

Step2: Check if price per ounce is the same

For \(8\) oz cup: \(\frac{2}{8}=0.25\)
For \(12\) oz cup: \(\frac{3}{12}=0.25\)
For \(16\) oz cup: \(\frac{4}{16}=0.25\)
For \(20\) oz cup: \(\frac{5}{20}=0.25\)

Step3: Interpret the slope in context

Since the slope \(m = 0.25=\frac{1}{4}\), it means for every \(4\) ounces increase in cup size, the price increases by \(1\) dollar. Equivalently, for every \(8\) ounces, \(\frac{8}{4}=2\) times the slope - related increment, so price increases by \(2\times1 = 2\) dollars. But if we consider the unit rate \(0.25=\frac{\text{Price increase}}{\text{Ounce increase}}\), cross - multiplying gives \(\text{Price increase}=0.25\times\text{Ounce increase}\). When \(\text{Ounce increase}=8\), \(\text{Price increase}=0.25\times8 = 2\) (this is an alternative way of looking at the proportion based on the slope value).

Answer:

1. Yes
2. \(0.25\)
3. For every \(8\) ounces of lemonade, the price increases by \(\$2\)