Power The formula for the power output P of a battery is P=V...

Power The formula for the power output $P$ of a battery is $P=V I-R I^{2}$ , where $V$ is the electromotive force in volts, $R$ is the resistance in ohms, and $I$ is the current in amperes. Find the current that corresponds to a maximum value of $P$ in a battery for which $V=12$ volts and $R=0.5$ ohm. Assume that a 15 -ampere fuse bounds the output in the interval $0 \leq I \leq 15 .$ Could the power output be increased by replacing the 15 -ampere fuse with a 20 -ampere fuse? Explain.

Key Concepts

Quadratic Function Optimization

The power output of the battery is expressed as a quadratic function in terms of current. Quadratic functions, characterized by a squared term, have a parabolic graph with a maximum (or minimum) point known as the vertex. This concept allows one to determine the current that maximizes power by either using the vertex formula or setting the derivative to zero.

Calculus in Optimization

Calculus plays a key role in optimization problems by enabling the determination of where a function reaches its maximum or minimum values. In this context, one calculates the derivative of the power function with respect to current and sets it equal to zero to find the optimal current that yields maximum power.

Electrical Circuit Analysis

Understanding the interplay between electromotive force (EMF), resistance, and current is fundamental in circuit analysis. The given power formula combines these elements, showing how internal resistance reduces the ideal power output provided by the battery's EMF, and thus highlights the practical limitations in electrical circuits.

Constraints in Optimization

Optimization in real-world scenarios often involves physical or safety constraints, such as a fuse's current rating. These constraints limit the range within which the theoretically optimal solution can be applied, ensuring that safety and operational guidelines are adhered to, even if the unconstrained mathematical optimum lies within the permissible range.

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