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Chapter 1: Problem 64

Would it ever be reasonable to use a quadratic model \(s(t)=a t^{2}+b t+c\) to predict long-term sales if \(a\) is negative? Explain.

Short Answer

Expert verified
Using a quadratic model with a negative \(a\) to predict long-term sales is not reasonable because it would result in the sales reaching a peak, then continuously decreasing over time. This is a pessimistic outlook, and a more realistic model would allow for growth followed by stability or consider other factors such as market saturation, competition, and product life cycles.

Step by step solution

01

Understand the components of the quadratic model

The quadratic model consists of three terms: \(at^2\), \(bt\), and \(c\). The term with the highest power, in this case, \(at^2\), determines the general shape and direction of the parabola. In this case, we are given that \(a\) is negative, which will affect the shape of the parabola.
02

Analyze the shape of the graph with a negative "a"

When the leading coefficient (in this case, \(a\)) of a quadratic equation is negative, the graph of the function is a downward-opening parabola. As time (\(t\)) goes by, the sales function (\(s(t)\)) will reach a maximum point and then start decreasing.
03

Consider the context of long-term sales

In the context of long-term sales, the assumption that sales grow indefinitely may not always be true, but it is also unlikely that sales will peak at a certain point and then steadily decrease. A downward-opening parabola, due to the negative leading coefficient (\(a\)), suggests that sales will continuously decrease after reaching the maximum point, which is a pessimistic outlook on long-term sales.
04

Make a conclusion

Since the model with a negative \(a\) predicts that sales will reach a peak then continuously decrease, it is not reasonable to use a quadratic model with a negative \(a\) for predicting long-term sales. A more realistic model might involve a function that allows for growth followed by stability, or more complex models that consider other factors such as market saturation, competition, and product life cycles.

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