Question

1. if you know the volume of something in pints, you can calculate its volume in gallons. complete the table below to this relationship, then answer the questions. pints, p ½ 4 12 16 32 gallons, g ½ a. what is the constant of proportionality? b. write an equation to represent the relationship between pints, p, and gallons, g. g = ____________

Explanation:

Part a: Constant of Proportionality

Step1: Recall Proportional Relationship

The relationship between pints (\(p\)) and gallons (\(g\)) is proportional, so \(g = kp\) where \(k\) is the constant of proportionality. We can use a pair of values from the table. Let's take \(p = 4\) and \(g=\frac{1}{2}\).

Step2: Solve for \(k\)

Substitute into \(g = kp\): \(\frac{1}{2}=k\times4\). Solve for \(k\): \(k=\frac{\frac{1}{2}}{4}=\frac{1}{8}\). Let's check with another pair, say \(p = 16\) and \(g = 2\) (from the table, when \(p = 16\), \(g = 2\)). \(k=\frac{2}{16}=\frac{1}{8}\). So the constant of proportionality is \(\frac{1}{8}\).

Step1: Use Proportional Formula

Since \(g\) and \(p\) are proportional with \(k=\frac{1}{8}\), the equation is \(g = kp\).

Step2: Substitute \(k\)

Substitute \(k=\frac{1}{8}\) into \(g = kp\), we get \(g=\frac{1}{8}p\).

Answer:

\(\frac{1}{8}\)

Part b: Equation for Relationship