The amount of waste a company produces, W, in tons per week,...

The amount of waste a company produces, $W$, in tons per week, is approximated by $W=3.75 e^{-0.008 t},$ where $t$ is in weeks since January $1,2016 .$ Waste removal for the company costs $\$ 150 /$ ton. How much did the company pay for waste removal during the year $2016 ?$

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Key Concepts

Exponential Decay

Exponential decay describes processes in which a quantity decreases at a rate proportional to its current value. In this context, the waste production decreases over time according to an exponential function, meaning that the rate of decline slows as time increases, which is typical for decay phenomena in various fields such as physics, biology, and economics.

Definite Integration

Definite integration involves calculating the total accumulation of a quantity over a specified interval. Here, integration is used to sum the continuously varying rate of waste production over the period of time in question, yielding the total amount of waste produced during that period.

Cost Calculation

Cost calculation in this problem involves determining the total expense by multiplying the total accumulated waste (obtained via integration) by the unit cost for waste removal. This step connects the mathematical model of waste production with the financial impact for practical decision making.

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