A country presently has coal reserves of 50 million tons....

A country presently has coal reserves of 50 million tons. Last year 6.5 million tons of coal was consumed. Past years' data and population projections suggest that the rate of consumption $R$ (in million tons/year) will increase according to the formula $R=6.5 e^{0.02 t},$ and the total amount $T$ (in million tons) of coal that will be used in $t$ years is given by the formula $T=325\left(e^{0.02 t}-1\right) .$ If the country uses only its own resources, when will the coal reserves be depleted?

Key Concepts

Exponential Growth

This concept involves processes that increase by a constant multiplicative rate over equal increments of time. It is characterized by functions in the form f(t) = a * e^(kt), where a is the initial quantity, k is the growth rate, and t is time. Exponential growth is common in many natural and social phenomena, such as population dynamics and compound interest, and requires careful analysis since it can lead to very rapid increases over time.

Integration of Exponential Functions

Integration is a fundamental calculus technique used to compute the total accumulation of a quantity when given its rate of change. Specifically, for exponential functions, the integral of an exponential function yields another exponential function scaled by the reciprocal of the coefficient in the exponent. This is essential for determining the total consumption or accumulation over a period when the underlying rate of change follows an exponential pattern.

Resource Depletion Analysis

This concept involves determining when a finite resource will be exhausted based on its rate of consumption. The analysis typically requires setting up a mathematical model where the available total resource is equated with the cumulative consumption over time. Solving this equation yields the time at which the resource will be depleted, providing key insights for resource management and planning.

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