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Exponential growth is a concept that describes a situation where a quantity increases at a consistent rate over time. This is often expressed with the formula: \text{Exponential Growth Formula: } \[ P(t) = P_0 \times \big(1 + \frac{r}{100}\big)^t \]In this expression,

• \textbf{\(P(t)\)} = the future value of the quantity
• \textbf{\(P_0\)} = the initial value
• \textbf{\(r\)} = the growth rate expressed as a percentage
• \textbf{\(t\)} = time period

By applying exponential growth to our oil production scenario, we see that every year the production increases by a rate of \(r\)percent. If we start with 4 billion barrels in the first year, after 10 years, the production will follow the pattern set by the exponential growth formula. This leads to the growth rate equation that we need to solve.

To solve equations, we often need to manipulate and test different values to check where a certain condition holds true. In our case, we need to solve: \[\big(1+\frac{r}{100}\big)^{10} - 0.15r - 1 = 0\] This kind of equation is not straightforward to solve algebraically, so we use a numerical approach, like a table, to find the value of \(r\):

• Start with a range of values, here \(8.00\) to \(8.15\), incrementing by \(0.05\).
• For each value, compute the left-hand side of the equation.

The correct value of \(r\) is found when the computed value is closest to zero. This value will give us the rate at which the oil production must increase annually to deplete the oil reserves over 10 years.

Oil reserves depletion refers to the gradual consumption of available oil resources. In this exercise, we look at an oil field with 60 billion barrels, annually producing 4 billion barrels, with production increasing annually at a rate \(r\). Over a span of 10 years, our goal is to find how quickly the production rate needs to increase to fully exhaust the reserves. The depletion rate is not constant; rather, it follows an exponential growth pattern due to compounding yearly production increases. Hence, we derive the expression \[ \big(1+\frac{r}{100}\big)^{10} - 0.15r - 1 = 0\] By adjusting \(r\), we match the scenario where the production over 10 years depletes all 60 billion barrels of reserves. This demonstrates not just the straightforward concept of depletion, but its interaction with growth rates and exponential functions.