A company that produces calculators estimates that the...

A company that produces calculators estimates that the profit $P$ (in dollars) from selling a specific model of calculator is given by $$P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45$$ where $x$ is the advertising expense (in tens of thousands of dollars). For this model of calculator, an advertising expense of $\$ 400,000(x=40)$ results in a profit of $\$ 2,174,375$. (a) Use a graphing utility to graph the profit function. (b) Use the graph from part (a) to estimate another amount the company can spend on advertising that results in the same profit. (c) Use synthetic division to confirm the result of part (b) algebraically.

A company that produces calculators estimates that the profit $P$ (in dollars) from selling a specific model of calculator is given by
$$P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45$$
where $x$ is the advertising expense (in tens of thousands of dollars). For this model of calculator, an advertising expense of $\$ 400,000(x=40)$ results in a profit of $\$ 2,174,375$.
(a) Use a graphing utility to graph the profit function.
(b) Use the graph from part (a) to estimate another amount the company can spend on advertising that results in the same profit.
(c) Use synthetic division to confirm the result of part (b) algebraically.

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Key Concepts

Equal Output Analysis

Equal output analysis involves finding different input values that result in the same value of a function. This is particularly useful in contexts like economics, where one might need to identify various conditions—such as different levels of spending—that yield the same profit or performance outcome.

Synthetic Division

Synthetic division is an efficient algorithm for dividing polynomials by linear factors of the form (x - c). It simplifies the process of polynomial division, making it easier to find roots, factor polynomials, and verify results obtained through other methods.

Polynomial Functions

Polynomial functions are expressions involving variables raised to non-negative integer exponents along with associated coefficients. They are fundamental in modeling various real-world scenarios, such as financial profit or cost, and their behavior—such as turning points and inflection points—can be analyzed to aid in decision making.

Graphing Utilities

Graphing utilities are software tools that enable users to visualize functions by plotting them on a coordinate plane. These visualizations help in understanding the behavior of functions, identifying key features like relative extrema, and estimating solutions, particularly when solving equations graphically.

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