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 $1,2005 .$ Waste removal for the company costs $\$ 15 /$ ton. How much did the company pay for waste removal during the year $2005 ?$

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Key Concepts

Exponential Decay

Exponential decay is a process where the quantity decreases at a rate proportional to its current value. In mathematical terms, it is often modeled by functions of the form A * e^(-kt), where A represents the initial amount, k is the decay constant, and t is time. This concept is applied in various fields to describe processes that diminish over time, such as radioactive decay, cooling, and the decreasing production of waste.

Definite Integration

Definite integration is used to compute the total accumulation of a continuously changing quantity over a specified interval. It involves calculating the area under the curve of a function between two limits. In this context, integration is essential to determine the total amount of waste produced over a specific time period by summing up the continuously varying waste production rate.

Cost Accumulation via Integration

Cost accumulation in problems involving continuous functions is achieved by integrating the rate function representing the quantity of interest and then applying a cost factor. Here, after integrating the waste production function over the given time interval, the resulting total waste is multiplied by the cost per unit to determine the overall expense for waste removal.

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