Question

To determine the time when sugar was cheapest and most expensive during the period \(1993 - 2003\).

Short answer

The time when sugar was cheapest is when \(t \approx 0.855\) (June 1994) and most expensive is when \(t \approx 4.618\) (March 1998) during the given period \(1993 - 2003\).

Step-by-step solution

Given information

The given function is \(s(t) = - 0.00003237{t^5} + 0.0009037{t^4} - 0.008956{t^3} + 0.03629{t^2} - 0.04458 + 0.4074\).

Definition of Maximum value

The local maximum is a point within an interval at which the function has a maximum value.

The absolute maxima is also called the global maxima and is the point across the entire domain of the given function, which gives the maximum value of the function.

Differentiate the function and find the Maximum values

Consider the given function\(s(t) = - 0.00003237{t^5} + 0.0009037{t^4} - 0.008956{t^3} + 0.03629{t^2} - 0.04458 + 0.4074\).

Differentiate the given function with respect to\(t\).

\({s^\prime }(t) = - 0.00016185{t^4} + 0.0036148{t^3} - 0.026868{t^2} + 0.07258t - 0.04458\)

Now, equate\({s^\prime }(t) = 0\)and evaluate the value of\(t\)by the use of a calculator.

\(\begin{aligned}{c}0 &= - 0.00016185{t^4} + 0.0036148{t^3} - 0.026868{t^2} + 0.07258t - 0.04458\\t &= 0.85478,4.6178,7.2919,9.5699\end{aligned}\)

The cheapest and the most expensive cost is evaluated by the substitution of the values of\(t\)obtained above in\(s(t) = 0\).

\(\begin{aligned}{c}s(0.85478) = 0.39068\\s(4.6178) = 0.43645\\s(7.2919) = 0.42712\\s(9.5699) = 0.43641\end{aligned}\)

From the above calculation it can be inferred that when\(t = 0.85478\)the expense of sugar is cheapest and when\(t = 4.6178\)the expense of sugar was expensive.

Here,\(0.85478 \approx 10\)months, that is\(10\)months from\(August{\rm{ }}1993\)is\(June{\rm{ }}1994\)and similarly,\(4.6178 \approx 4\)years 7 months, that is 4 years 7 months from\(August{\rm{ }}1993\)is\(March{\rm{ }}1998.\)

Therefore, the time when sugar was cheapest is when \(t \approx 0.855\) (\(June{\rm{ }}1994\)) and most expensive is when \(t \approx 4.618\)(March 1998) during the given period\(1993 - 2003\).