A state legislature is debating two proposals for...

A state legislature is debating two proposals for eliminating the annual budget deficits by the year $2010 .$ The rate of decrease of the deficits for each proposal is shown in the figure. From the viewpoint of minimizing the cumulative state deficit, which is the better proposal? Explain.

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Key Concepts

Timing of Reductions

This concept underlines the importance of the temporal distribution of the rate of decrease. When deficit reductions occur earlier in the timeline, the cumulative deficit is lower compared to if the reductions occur later. Evaluating proposals on when the rates are highest—earlier rather than later—helps determine which strategy minimizes the overall accumulated deficit.

Rate of Change

This concept refers to how quickly a quantity—in this case, the state deficit—is changing over time. Understanding the rate at which the deficit decreases is crucial because it indicates how rapidly the accumulated deficits are being reduced. A higher rate of reduction early on helps in diminishing the overall accumulated debt more effectively.

Integral of a Rate Function

Integrating a rate function over a period of time provides the cumulative effect of that rate. In the context of the state deficit, the integral of the rate of decrease over time gives the total reduction in the deficit. This integration is key to calculating the entire amount saved over the period and hence deciding which proposal minimizes the cumulative deficit better.

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