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 in Ghana decreasing?

Short answer

Between the years 1980 and 1984, the production of gold in Ghana was decreasing.

Step-by-step solution

Find the derivative

Calculate the derivative of the function G, using the power rule, which states that the derivative of \(ax^n\) is \(anx^{n-1}\). So, for our function, of which the derivative is noted as G', it will be \(G' = 24t - 103\).

Find the turning points

Set the derivative equal to zero, and solve for \(t\). \(24t - 103 = 0\). Solving for \(t\) gives approximately \(t \approx 4.29\) years after 1980, so around the year 1984.

Test intervals around turning points

The function usually changes its behavior at turning points. Therefore, pick two values, one from 1980 to 1984 and other from 1984 to 1995, and evaluate the derivative at those points. Let's pick t=2 (1982) and t=6 (1986). \(G'(2) = -55<0\) and \(G'(6) = 41>0\).

Determine decreasing intervals

Since the derivative at t=2 (1982) is less than zero, this means the function was decreasing in that interval from 1980 to 1984. Then it started to increase as indicated by a positive derivative when evaluated at year 1986.