Question

1. shawn charges $15 for each lawn he mows. is this a proportional relationship? if so, determine the constant of proportionality.
2. given the table, determine whether the amount of money earned is proportional to the number of hours worked. if so, calculate the constant of proportionality and explain what it means in this situation.

time worked (hours) 0.5 4 5.5 10
money earned (dollars) 3.25 26 35.75 65

Explanation:

Step1: Recall the definition of proportionality

Two variables are proportional if the ratio of their values is constant. Let $x$ be the time worked and $y$ be the money earned. We need to check if $\frac{y}{x}$ is the same for all pairs of values.

Step2: Calculate the ratios for each pair

For the first pair: $x = 0.5$, $y=3.25$, then $\frac{y}{x}=\frac{3.25}{0.5}=6.5$.
For the second pair: $x = 4$, $y = 26$, then $\frac{y}{x}=\frac{26}{4}=6.5$.
For the third pair: $x = 5.5$, $y=35.75$, then $\frac{y}{x}=\frac{35.75}{5.5}=6.5$.
For the fourth pair: $x = 10$, $y = 65$, then $\frac{y}{x}=\frac{65}{10}=6.5$.

Step3: Determine if it's proportional

Since the ratio $\frac{y}{x}=6.5$ for all pairs of time - worked and money - earned values, the amount of money earned is proportional to the number of hours worked. The constant of proportionality is $6.5$. This means that for every 1 - hour increase in the time worked, the money earned increases by $6.5$ dollars.

Answer:

Yes, the amount of money earned is proportional to the number of hours worked. The constant of proportionality is $6.5$. It means that for each hour of work, $6.5$ dollars are earned.