Question

the table below shows fengs earnings on the job.
time (hours) | earnings (dollars)
21 | $552.30
23 | $604.90
31 | $815.30
if e represents his total earnings in dollars and cents for any number of hours worked, h, write a proportional equation for e in terms of h that matches the context.

Explanation:

Step1: Find the rate of earnings (slope)

To find the rate, we can use the formula for slope (rate) \( r=\frac{\Delta e}{\Delta h} \). Let's take the first two points \((21, 552.30)\) and \((23, 604.90)\). The change in earnings \(\Delta e = 604.90 - 552.30=52.60\) and the change in hours \(\Delta h=23 - 21 = 2\). So the rate \( r=\frac{52.60}{2}=26.3 \). We can verify with another pair, say \((21,552.30)\) and \((31,815.30)\). \(\Delta e=815.30 - 552.30 = 263\), \(\Delta h=31 - 21=10\), and \(\frac{263}{10}=26.3\). So the rate is \(26.3\) dollars per hour.

Step2: Write the proportional equation

Since earnings \( e \) are proportional to hours \( h \) (because the rate is constant), the equation is \( e = r\times h \), where \( r = 26.3 \). So \( e=26.3h \).

Answer:

\( e = 26.3h \)