- Manuel's total earnings for 7 hours of work is \(\$56\).
- The algebraic expression for Manuel's earnings after \(t\) hours is \(E = 8t\).
- Using an algebraic expression allows for quick, generalized calculations of income for any number of hours worked.
- Final answers: a. \(\boxed{\$56}\), b. \(\boxed{8t}\), c. Provides a functional model for rapid calculation.
Explanation
- Calculating Earnings for 7 Hours
To find Manuel's total earnings for his work this week, we identify the hourly rate and the number of hours worked. Manuel earns \(\$8\) per hour and has worked \(7\) hours. By multiplying the hourly rate by the number of hours, we calculate his total earnings:
\[8 \text{ dollars/hour} \times 7 \text{ hours} = \boxed{\$56}\]
- Developing an Algebraic Expression
To generalize Manuel's earnings for any number of hours worked, we represent the number of hours as a variable \(t\). Since he makes \(\$8\) for every hour worked, the total amount of money earned, let's call it \(E\), is the product of his hourly rate and the number of hours. Thus, the algebraic expression for his earnings is:
\[E = 8t\]
- Explaining the Utility of the Expression
Writing an algebraic expression for Manuel's earnings is beneficial because it creates a functional model that simplifies the calculation process for any given duration of time. Instead of repeatedly performing the same multiplication, Manuel can simply substitute any value of \(t\) into the expression \(8t\) to determine his earnings instantly. This provides a versatile tool for planning his income based on different work schedules or tracking earnings over various periods.
Examples
Algebraic expressions are used in everyday life to calculate costs, income, or time. For instance, if you are buying items at a store where each item costs the same amount, you can use an expression like \(C = n \times P\), where \(C\) is the total cost, \(n\) is the number of items, and \(P\) is the price per item. This allows you to quickly figure out how much you need to pay, whether you are buying 1 item or 100 items, just by changing the value of \(n\).