A two-day environmental cleanup started at 9 am on the first...

A two-day environmental cleanup started at 9 am on the first day. The number of workers fluctuated as shown in Figure $5.61 .$ If the workers were paid $\$ 10$ per hour, how much was the total personnel cost of the cleanup?
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Key Concepts

Piecewise Functions

This concept involves functions that are defined by different expressions or values over different intervals. In many practical situations, such as tracking the number of workers over time, the value changes at discrete intervals. Understanding piecewise functions enables the segmentation of the overall problem into manageable parts where each time segment can be analyzed separately.

Integration and Area Under a Curve

Integration, especially when viewing it as the area under a curve, is key to summing values over a continuous interval. In contexts where a quantity (like the number of workers) varies over time, calculating the total 'area' under the curve provides an aggregate measure, such as total man-hours worked, by summing contributions from each interval.

Unit Conversion and Cost Calculation

After calculating a total quantity like man-hours from an integrated or summed value, the final step is converting these units into a cost. This involves multiplying the total time units by the rate per unit of labor. Understanding unit conversion ensures that the appropriate mathematical operations are performed to arrive at accurate real-world cost estimates.

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