Question

In Exercises \(69-71\), use the following information. In Ghana from 1980 to \(1995,\) the annual production of gold \(G\) in thousands of ounces can be modeled by \(G=12 t^{2}-103 t+434,\) where \(t\) is the number of years since 1980 . From 1980 to \(1995,\) during which years was the production of gold increasing?

Short answer

The gold production in Ghana was increasing from the years 1985 through 1995.

Step-by-step solution

Find the Derivative of G(t)

Determine the derivative of \(G(t) = 12t^2 - 103t + 434\), denoted as \(G'(t)\). Apply the power rule: the derivative of \(t^2\) is \(2t\), and the derivative of \(t\) is \(1\). So, \(G'(t) = 24t - 103\).

Set G'(t) > 0

For the production of gold to be increasing, the rate of change of production with respect to time (represented by \(G'(t)\)) should be positive. Therefore, set \(G'(t) > 0\). You should have the inequality \(24t - 103 > 0\).

Solve for t

Solve the inequality for \(t\). Add 103 to both sides to obtain \(24t > 103\). Divide both sides by 24 to find \(t > 103 / 24 \approx 4.29\).

Interpret the Result Correctly

The resulting value of \(t\) should represent years since 1980. A value of \(t > 4.29\) indicates that gold production was increasing starting from a little over 4 years into the time period, or from roughly late 1984. Since the time period ends in 1995, it can be concluded that gold production increased during the years 1985 through 1995.

Key concepts

Derivative of a Function

Understanding the derivative of a function is crucial for analyzing real-world situations where change is involved. In the context of increasing gold production, the derivative represents the rate at which gold production is changing over time. The power rule is a handy tool to find derivatives quickly. For any term in the form of \(at^n\), the derivative is \(ant^{n-1}\). If the exponent is 2 as in \(t^2\), then the derivative is \(2t\). And for any term that is simply \(t\), it becomes a constant upon differentiation.

Applying the power rule to the function \(G(t) = 12t^2 - 103t + 434\), we arrive at its derivative \(G'(t) = 24t - 103\), which provides the basis for determining when production is increasing. It simplifies our understanding of the problem by representing a complex phenomenon with a single formula.

Solving Inequalities

In the world of algebra, inequalities tell us about the relative size of two expressions. Solving inequalities is pivotal for answering questions about ranges such as 'during which years was production increasing?' In our problem, the inequality \(24t - 103 > 0\) emerges from the derivative, indicating a positive rate of change.

To handle such inequalities, apply a step-by-step approach: isolate the variable \(t\) by performing the same operation on both sides of the inequality. Adding 103 to both sides and then dividing by 24, essentially 'unwraps' the inequality to solve for \(t\). Finally, interpreting the inequality in the context of the scenario gives us the answer: gold production was increasing when \(t > 4.29\). It's through such inequalities that algebra connects to practical situations, giving clear-cut answers.

Applied Algebra in Real-World Context

Algebra isn't just about abstract equations and variables; it's a powerful tool for modeling and solving real-world problems. The function \(G(t) = 12t^2 - 103t + 434\), representing gold production in Ghana, is an instance of applied algebra. In this model, \(t\) is a real-world quantity — the number of years since 1980, and \(G(t)\) is the production in thousands of ounces. The algebraic model enables us to project, analyze, and make decisions based on the production trend, which is incredibly valuable for industries and economists alike.

Using such a model to find when the production of gold is increasing brings abstract mathematics down to a tangible level. It turns the general concept of derivatives and inequalities into a distinct, applicable finding, reinforcing the practical importance of algebra.

Rate of Change

The rate of change is a universal concept observed across various disciplines. It's a measure of how a quantity, such as gold production, changes over time. In the algebra problem we're discussing, the derivative \(G'(t) = 24t - 103\) gives us this rate. When the derivative is positive, our rate of change is positive, indicating that the quantity is increasing.

We seek this positive increase when solving for \(t\) — the point in time after which gold production is on the upswing. The understanding of rate of change is not only fundamental to mathematics but is also crucial in fields such as physics, economics, and engineering, underpinning countless phenomena and systems in our world.