Piecewise linear functions

Question

Derek earns $25 per hour (or part of an hour) for a regular 40-hour work week. Sometimes, he works over 40 hours a week and earns “double time” for the extra hours worked. (That means that his hourly rate for the additional hours over 40 hours per week is double his normal hourly rate.) Per work regulations, Derek can work at most 50 hours a week. 

A) If Derek works 46 hours in one week, how much would he earn that week? (Make sure you include units)

B) Create a piecewise linear function to represent the amount Derek will earn in one week if he works xxhours. (Make sure to define your variables and use proper notation)

C) Graph this function in an appropriate window. (Make sure to label your axes, scale and any important points.) 

D) Is your graph continuous?

Answer 1

Given:

Derek earns $25 per hour for the first 40 hours and earn double for the extra hours worked.

A) Calculating how much Derek would make if he worked 46 hours:

For the first 40 hours, the rate is $25 per hour so we can do:

40 x 25 = $1000

For the hours work after 40 hours are extra hours which is 6 hours since 46 - 40 = 6 hours. For extra hours, Derek earns double his regular rate which is $25 so he makes $50 for his extra hours:

6 x 50 = $300

Now, we can add them up to find out how much total did Derek make in 46 hours:

$1000 + $300 = $1300

Derek would earn $1300 that week.

B) Creating a piecewise linear function to represent the amount Derek will earn in one week if he works x hours:

Let x represent the number of hours Derek works in one week, and let E(x) represent Derek's weekly earnings in dollars. So, our piecewise function would be:

E(x) = {25x, 0 ≤ x ≤ 40

{1000 + 50(x-40), 40 < x ≤ 50

You can simplify the second part so it becomes:

E(x) = {25x, 0 ≤ x ≤ 40

{50x - 1000, 40 < x ≤ 50

In this piecewise function, the first sub-function represents Derek's earnings rate for the first 40 hours which will be $25 per hour so 25x. The second sub-function represents Derek's earnings after 40 hours. He already earned $1000 (40 x 25 = 1000) for the first 40 hours, and then he earns $50 per hour only for the extra hours after 40.

C) Graphing the function:

I am unable to graph here but you can use the window:

0 ≤ x ≤ 50 and 0 ≤ y ≤ 1500.

The graph starts at (0, 0), goes up linearly to (40, 1000), then becomes steeper from (40, 1000) to (50, 1500) because Derek earns double time after 40 hours.

D) Is the graph continuous?

Yes, the graph is continuous because there is no break, gap, or jump at x = 40. Both pieces meet at the point (40, 1000). We can test this by plugging 40 into both sub-functions of the piecewise function:

First sub-function:

E(x) = 25x

E(40) = 25(40) = 1000

Second sub-function:

E(x) = 50x - 1000

E(40) = 50(40) - 1000

= 2000 - 1000

= 1000

Notice, both pieces give the same value (1000) at x = 40. Therefore, the graph meets at the point (40, 1000) and so graph is continuous since there is no gap, hole, or jump at x = 40.

Answer 2

a)46=40+6

40*$25+6*(2*$25)=$25*(40+12)=$25*52=$1300

b) y=40*$25+x*(2*$25)= 50x+1000, y=50x+1000