The relationship between the amount of money x that Cannon...

The relationship between the amount of money $x$ that Cannon Precision Instruments spends on advertising and the company's total sales $S(x)$ is given by the function $$ S(x)=-0.002 x^{3}+0.6 x^{2}+x+500 \quad(0 \leq x \leq 200) $$ where $x$ is measured in thousands of dollars. Find the rate of change of the sales with respect to the amount of money spent on advertising. Are Cannon's total sales increasing at a faster rate when the amount of money spent on advertising is (a) $\$ 100,000$ or (b) $\$ 150,000 ?$

The relationship between the amount of money $x$ that Cannon Precision Instruments spends on advertising and the company's total sales $S(x)$ is given by the function
$$
S(x)=-0.002 x^{3}+0.6 x^{2}+x+500 \quad(0 \leq x \leq 200)
$$
where $x$ is measured in thousands of dollars. Find the rate of change of the sales with respect to the amount of money spent on advertising. Are Cannon's total sales increasing at a faster rate when the amount of money spent on advertising is
(a) $\$ 100,000$ or
(b) $\$ 150,000 ?$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which defines the instantaneous rate of change of the function with respect to its variable. In the context of the problem, finding the derivative of the sales function with respect to advertising spending gives insight into how sales are changing for a given incremental increase in advertising dollars.

Instantaneous Rate of Change

The instantaneous rate of change, represented by the derivative, tells us how rapidly the dependent variable (sales) is changing at a specific point of the independent variable (advertising spending). By evaluating this rate at different spending levels, one can determine which level yields a higher growth rate in sales.

Interpreting Derivatives in Business Context

In business and economics, derivatives are used to analyze the sensitivity and responsiveness of one variable, such as sales, with respect to changes in another variable, such as advertising expenditures. This analysis helps in making informed decisions about the optimal allocation of resources to maximize growth or returns.

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