The table shows how much a carpenter charges for work. Is the relationship shown by the data in the table linear? Explain your answer. Hours Worked: 1,2,3,4
Amount Charged ($) 25, 40, 60, 0
To determine if the relationship between "Hours Worked" and "Amount Charged" is linear, we look at the rate of change between the pairs of data points. In a linear relationship, the rate of change (or the slope, in the case of a graph) between any two points should be constant.
Let's calculate the differences in the "Amount Charged" for each additional hour worked:
1. For the first two data points:
When going from 1 to 2 hours, the Amount Charged goes from $25 to $40. That's an increase of $15 for 1 additional hour of work.
2. For the second and third data points:
When going from 2 to 3 hours, the Amount Charged goes from $40 to $60. That's an increase of $20 for 1 additional hour of work.
3. For the third and fourth data points:
When going from 3 to 4 hours, the Amount Charged goes from $60 to $0. This is a decrease of $60, which is peculiar and may indicate either an error in the data or there might be special pricing conditions not indicated by the data in the table.
The differences in the amounts are not consistent. We see an increase of $15, then $20, and then a decrease of $60 (which is problematic as it results in a negative charge, which doesn't make sense in a real-world scenario).
Given these differences, it is clear that the relationship is not linear. Moreover, the last data point with 0 dollars charged for 4 hours worked either indicates a mistake or that this is not a simple hourly rate payment system. A linear relationship would need the amount charged to continuously increase by the same amount for each hour worked without such erratic changes.