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) | money earned (dollars)
0.5 | 3.25
4 | 26
5.5 | 35.75
10 | 65

Explanation:

Problem 1

Step1: Recall proportional relationship

A proportional relationship has the form \( y = kx \), where \( k \) is the constant of proportionality. For Shawn, let \( x \) be the number of lawns and \( y \) be the money earned. He charges $15 per lawn, so \( y = 15x \).

Step2: Check proportionality

Since it's in the form \( y = kx \) (no initial fee, just per lawn), it's proportional. The constant \( k \) is 15.

Step1: Calculate ratio for each pair

For a proportional relationship, \( \frac{\text{Money Earned}}{\text{Time Worked}} \) should be constant.

• For \( 0.5 \) hours and $3.25: \( \frac{3.25}{0.5} = 6.5 \)
• For \( 4 \) hours and $26: \( \frac{26}{4} = 6.5 \)
• For \( 5.5 \) hours and $35.75: \( \frac{35.75}{5.5} = 6.5 \)
• For \( 10 \) hours and $65: \( \frac{65}{10} = 6.5 \)

Step2: Determine proportionality and constant

All ratios are 6.5, so it's proportional. The constant \( k = 6.5 \) means the hourly wage is $6.50 (money earned per hour worked).

Answer:

Yes, it is a proportional relationship. The constant of proportionality is 15 (dollars per lawn).

Problem 2