V. King Manufacturing buys a new machine for 250,000...

V. King Manufacturing buys a new machine for 250,000 dollars. The marginal revenue from the sale of products produced by the machine after $t$ years is given by $R^{\prime}(t)=4000 t$ The salvage value of the machine, in dollars, after $t$ years is given by $V(t)=200,000-25,000 e^{0.1 t}$ The total profit from the machine, in dollars, after $t$ years is given by $$P(t)=\left(\begin{array}{c}\text { Revenue } \\\text { from } \\\text { sale of } \\\text { product } \end{array}\right)+\left(\begin{array}{c}\text { Revenue } \\\text { from } \\\text { sale of } \\\text { machine }\end{array}\right)-\left(\begin{array}{c}\text { cost } \\\text { of } \\\text { machine } \end{array}\right)$$ The company knows that $R(0)=0$ a) Find $P(t)$ b) Find $P(10)$

V. King Manufacturing buys a new machine for 250,000 dollars. The marginal revenue from the sale of products produced by the machine after $t$ years is given by $R^{\prime}(t)=4000 t$ The salvage value of the machine, in dollars, after $t$ years is given by $V(t)=200,000-25,000 e^{0.1 t}$ The total profit from the machine, in dollars, after $t$ years is given by
$$P(t)=\left(\begin{array}{c}\text { Revenue } \\\text { from } \\\text { sale of } \\\text { product }
\end{array}\right)+\left(\begin{array}{c}\text { Revenue } \\\text { from } \\\text { sale of } \\\text { machine }\end{array}\right)-\left(\begin{array}{c}\text { cost } \\\text { of } \\\text { machine }
\end{array}\right)$$
The company knows that $R(0)=0$
a) Find $P(t)$
b) Find $P(10)$

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Key Concepts

Initial Condition Application

When integrating a rate function to recover an accumulated quantity, initial conditions are essential to solve for the constant of integration. In this context, knowing the revenue at time zero ensures that the revenue function accurately reflects the system's behavior from its starting point.

Profit Function Formulation

The profit function is constructed by combining all sources of revenue, such as earnings from operations and salvage proceeds, and then subtracting the associated costs. This comprehensive approach allows for the accurate evaluation of the financial viability and return on investments in assets.

Salvage Value

Salvage value represents the residual or remaining value of an asset at a given time when it is sold or disposed of. It is a critical factor in asset valuation and cost-benefit analyses, as it influences the overall profit calculation by providing an additional revenue stream at the asset's end of life.

Marginal Revenue and Integration

This concept deals with the interpretation of a derivative (marginal revenue) as the instantaneous rate of change of a quantity (total revenue). Integrating the marginal revenue function over time provides the accumulated total revenue, with an integration constant determined by initial conditions.

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