Determine the function w = f(n), where w equals the hourly wage rate in dollars and n equals the number of hours required to complete one unit of the product.
Understand the Problem
The question involves determining a function for the hourly wage rate based on the number of hours worked to complete a unit of a product, including an incentive for completing the unit in less than a specified time. It outlines the incremental increase in wages for each hour under the 15-hour standard and provides an example calculation.
Answer
The hourly wage function is $w(n) = 30.5 - 1.50n$.
Answer for screen readers
The function for the hourly wage $w$ based on the number of hours $n$ worked is: $$ w(n) = 30.5 - 1.50n $$
Steps to Solve
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Define the basic wage rate The base hourly wage is given as $8.00. This will be the wage used before applying the incentive for hours worked.
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Identify the standard hours and incentives The standard hours for completion are 15 hours. For each hour less than this standard, the incentive is $1.50 per hour.
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Calculate the number of hours under the standard If a worker completes a unit in $n$ hours, the number of hours saved can be calculated as: $$ \text{Hours saved} = 15 - n $$
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Determine the incentive applied based on hours saved The incentive for completing the work in less than 15 hours can be calculated as: $$ \text{Incentive} = (15 - n) \cdot 1.50 $$
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Construct the function for the hourly wage To express the hourly wage $w$ as a function of $n$, include the base wage and the incentive: $$ w(n) = 8 + (15 - n) \cdot 1.50 $$
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Simplify the function Distributing the incentive yields: $$ w(n) = 8 + 22.5 - 1.50n $$
Thus, $$ w(n) = 30.5 - 1.50n $$
The function for the hourly wage $w$ based on the number of hours $n$ worked is: $$ w(n) = 30.5 - 1.50n $$
More Information
This function indicates that for every hour worked beyond 15 hours, the hourly wage decreases by $1.50. If a unit is completed at 15 hours, the worker earns $8.00. Completing the unit faster increases the worker's wage due to the incentive.
Tips
- Ignoring the base wage: Always ensure the base wage is included when calculating the total.
- Not accounting for fractions of hours: Remember to apply the incentive even if the number of hours is in fractions.
- Miscalculating the incentive: Ensure that the incentive is calculated based on $15 - n$ consistently.
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