A computer technician keeps track of his earnings throughout each month. the technician observes that his earnings are a linear function of the number of hours he works during the month. The technician finds that when he works 55 hours during the month, he earn $2,125 and when he works 30 hours, he earns $585

Part A: Write a linear function to model the relationship between the number of hours worked and the money earned.

Function _____________

Part B: Explain the meaning of slope in the context of the problem.

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Part A: To write a linear function that models the relationship between the number of hours worked and the money earned, we need to find the slope (m) and the y-intercept (b) of the line.

We can use the formula for the equation of a line in slope-intercept form: y = mx + b, where y is the dependent variable (earnings in this case), x is the independent variable (number of hours worked), m is the slope, and b is the y-intercept.

Given that the technician earns $2,125 when working 55 hours and $585 when working 30 hours, we can use these two points to find the slope:

slope (m) = (change in y) / (change in x)
= (2125 - 585) / (55 - 30)
= 1540 / 25
= 61.6

Now that we have the slope (m), we can use either of the given points to find the y-intercept:
Using the point (55, 2125):
y = mx + b
2125 = 61.6(55) + b
2125 = 3386 + b
b = 2125 - 3386
b = -1261

Therefore, the linear function that models the relationship between the number of hours worked (x) and the money earned (y) is:
y = 61.6x - 1261

Part B: In this problem, the slope (m) represents the rate at which the technician's earnings increase or decrease with each additional hour worked. It indicates how much additional money the technician earns for each additional hour worked.

In this case, the slope of 61.6 means that for every additional hour the technician works, his earnings increase by $61.6. This indicates a positive correlation between the number of hours worked and the money earned.

A higher slope value would mean a steeper increase in earnings with each additional hour, while a lower slope value would indicate a more gradual increase. In general, the slope quantifies the relationship between the independent and dependent variables in a linear function.

Part A: Write a linear function to model the relationship between the number of hours worked and the money earned.

Let's assume the number of hours worked per month as x and the earnings as y. Given that the earnings are a linear function of the number of hours worked, we can write the linear function as:

y = mx + b

To find the values of m and b, we can use the information provided.

The technician earns $2,125 when he works 55 hours, so we have the point (55, 2125).

The technician earns $585 when he works 30 hours, so we have the point (30, 585).

Using these two points, we can find the values of m and b.

Using the formula for slope (m):

m = (y2 - y1) / (x2 - x1)

m = (2125 - 585) / (55 - 30)

m = 1540 / 25

m = 61.6

So, the slope of the linear function is 61.6.

Using the formula for the y-intercept (b):

b = y - mx

Using the point (55, 2125):

b = 2125 - (61.6 * 55)

b = 2125 - 3398

b = -1273

So, the y-intercept of the linear function is -1273.

Therefore, the linear function that models the relationship between the number of hours worked and the money earned is:

y = 61.6x - 1273

Part B: Explain the meaning of slope in the context of the problem.

In the context of this problem, the slope (61.6) represents the amount of money earned for each additional hour worked. It tells us the rate of increase or decrease in earnings per hour. In this case, for every hour the technician works, he earns an additional $61.6. So, the slope represents the earnings per hour.