Question

question 17 of 25
an engineering technician makes $25 an hour for the first 40 hours she works during a week and $32 an hour for each hour over 40 hours. which piecewise equation models her total weekly pay y in dollars as it relates to the number of hours x that she has worked during the week?

a. $y = \

$$\begin{cases} 25x & 0 \\leq x \\leq 40 \\\\ 32x & x > 40 \\end{cases}$$

$

b. $y = \

$$\begin{cases} 25x & 0 \\leq x \\leq 40 \\\\ 32(x - 40) & x > 40 \\end{cases}$$

$

c. $y = \

$$\begin{cases} 25x & 0 \\leq x \\leq 40 \\\\ 32x + 1000 & x > 40 \\end{cases}$$

$

d. $y = \

$$\begin{cases} 25x & 0 \\leq x \\leq 40 \\\\ 32(x - 40) + 1000 & x > 40 \\end{cases}$$

$

Explanation:

Step1: Analyze pay for \( 0 \leq x \leq 40 \)

For the first 40 hours, the rate is $25 per hour. So pay \( y = 25x \) (since total pay = rate × hours).

Step2: Analyze pay for \( x > 40 \)

For hours over 40, she gets $25 for the first 40 hours (total LXI0) and $32 for each additional hour (\( x - 40 \)). So total pay \( y = 32(x - 40) + 1000 \).

Step3: Evaluate options

• Option A: Wrong, as over 40 hours should include the first 40 hours' pay, not just \( 32x \).
• Option B: Wrong, as it doesn't include the $1000 from the first 40 hours.
• Option C: Wrong, the over - 40 formula is incorrect (should be based on additional hours, not \( 32x+1000 \)).
• Option D: Correct, matches the derived formulas for both intervals.

Answer:

D. \( y =

$$\begin{cases} 25x & 0 \leq x \leq 40 \\ 32(x - 40) + 1000 & x > 40 \end{cases}$$

\)