Math Problem Statement

A construction worker earned $17 per hour for the first 40 hours of work and $25.50 per hour for work in excess of 40 hours. (a) Write a function to model her weekly pay W in terms of the number of hours she worked that week. (b) One week she earned $896.75. How much overtime did she work?

Solution

Let's break down the problem step by step.

(a) Write a function for weekly pay WW:

The construction worker earns:

  • $17 per hour for the first 40 hours.
  • $25.50 per hour for any hours worked beyond 40 hours.

Let hh be the total number of hours worked in a week.

The function for the weekly pay W(h)W(h) would be:

  • If h≤40h \leq 40, then W(h)=17hW(h) = 17h (since no overtime is worked).
  • If h>40h > 40, then W(h)=17×40+25.5×(h−40)W(h) = 17 \times 40 + 25.5 \times (h - 40).

So, the function can be written as:

\begin{cases} 17h & \text{if } h \leq 40 \\ 17 \times 40 + 25.5 \times (h - 40) & \text{if } h > 40 \end{cases}$$ ### (b) She earned $896.75 in one week. How much overtime did she work? To solve this, we know her total earnings, so we can set up the equation using the function $$W(h)$$ and solve for $$h$$. If she worked overtime, the equation will be: $$896.75 = 17 \times 40 + 25.5 \times (h - 40)$$ Let me calculate the value of $$h$$ from this equation. The total number of hours worked was 48.5 hours. Since she worked 48.5 hours and the first 40 hours are regular hours, the overtime hours worked are: $$48.5 - 40 = 8.5 \text{ hours}.$$ Thus, she worked **8.5 hours** of overtime. --- Do you want further details on this, or have any questions? Here are 5 related questions: 1. How would the function change if the overtime pay rate increased to $30 per hour? 2. What would her weekly pay be if she worked 50 hours in a week? 3. How much would she earn for 35 hours of work? 4. What is the overtime rate as a percentage of her regular pay rate? 5. If she worked 60 hours, how much would be earned from overtime alone? **Tip**: When working with piecewise functions, clearly define the conditions and calculate each part separately for accuracy.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Algebra

Formulas

W(h) = 17h for h ≤ 40
W(h) = 17 * 40 + 25.5 * (h - 40) for h > 40

Theorems

-

Suitable Grade Level

Grade 10-12

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