Problem 108 - Algebra for College Students

Question

If Mildred earns \(\$ 14.50\) per hour, express her total pay \(P(h)\) as a function of the number of hours worked \(h .\) Find \(P(40)\).

Step-by-Step Solution

Answer

P(40) = 580.00

Step 1 - Understand the Problem

Identify that Mildred's pay rate is given and we need to express her total pay as a function of hours worked.

Step 2 - Define the Function

Let the total pay be represented by the function P(h). Mildred earns $14.50 per hour, so the total pay is the hourly rate multiplied by the number of hours worked. Therefore, the function is: \[ P(h) = 14.50 \times h \]

Step 3 - Substitute the Given Hours

To find Mildred's pay for 40 hours of work, substitute \(h = 40\) into the function: \[ P(40) = 14.50 \times 40 \]

Step 4 - Calculate the Total Pay

Now perform the multiplication to find the total pay: \[ P(40) = 580.00 \]

Key Concepts

Linear Functions

Linear functions are a type of function that create a straight line when graphed on a coordinate plane. They have the general form y = mx + b, where:- y is the dependent variable (output)- x is the independent variable (input)- m is the slope (the rate of change)- b is the y-intercept (the value of y when x is zero) In the exercise, Mildred's earnings can be represented as a linear function because her pay increases at a constant rate of $14.50 per hour. This rate (14.50) is akin to the slope in the general equation, and there is no y-intercept in this case since her pay is zero when she does not work any hours.

Function Notation

Function notation is a way to represent functions in a more compact form. It typically looks like this: f(x), where 'f' is the name of the function and 'x' is the input variable. In the given exercise, the function is named P(h) to signify the total pay P as a function of the hours worked, h.
Using function notation, we can effectively express Mildred's pay equation as: P(h) = 14.50h This notation is very useful because it allows us to quickly find the output for any input. For example, to find the pay for 40 hours of work, substitute 40 for h in the equation: P(40) = 14.50 × 40.

Real-World Applications

Understanding linear functions and function notation has practical applications in many real-life scenarios. For instance, in the given exercise, determining Mildred's total pay based on her hourly rate is a straightforward application of a linear function. Such functions can be used for various calculations, including:

• Budgeting: Predicting expenses based on utility rates or subscription fees.
• Time management: Estimating the duration required to complete a task at a steady pace.
• Business: Calculating profits, losses, and break-even points.

Using function notation makes it easier to implement these calculations in algebra, spreadsheet software, and even coding. Mastering these concepts can significantly enhance financial planning and resource management.