A company that produces calculators estimated that the...

A company that produces calculators estimated that the profit $P$ (in dollars) from selling a particular model of calculator was $P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45$ where $x$ was the advertising expense $($ in tens of thousands of dollars). For this model of calculator, the advertising expense was $\$ 400,000(x=40)$ and the profit was $\$ 2,174,375.$ (a) Use a graphing utility to graph the profit function. (b) Use the graph from part (a) to estimate another amount the company could have spent on advertising that would have produced the same profit. (c) Use synthetic division to confirm the result of part (b) algebraically.

A company that produces calculators estimated that the profit $P$ (in dollars) from selling a particular model of calculator was
$P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45$
where $x$ was the advertising expense $($ in tens of thousands of dollars). For this model of calculator, the advertising expense was $\$ 400,000(x=40)$ and the profit was $\$ 2,174,375.$
(a) Use a graphing utility to graph the profit function.
(b) Use the graph from part (a) to estimate another amount the company could have spent on advertising that would have produced the same profit.
(c) Use synthetic division to confirm the result of part (b) algebraically.

Key Concepts

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a binomial of the form (x - c). It provides a quicker and simpler way to check whether a given value is a root of the polynomial and to factor the polynomial, thereby confirming solutions suggested by other methods such as graphing.

Polynomial Function Modeling

This concept involves representing real-world relationships using polynomial equations. In such models, quantities like profit, cost, or revenue are expressed in terms of an independent variable, often leading to equations whose coefficients and degrees capture the nuances of how different factors interact.

Graphing Polynomial Functions

Graphing is a vital tool for analyzing the behavior of polynomial functions. It helps in visualizing the function's overall shape, its turning points, intercepts, and trends across a given domain. Such a visual approach is essential for understanding and interpreting the function’s properties and for estimating key values.

Estimating Solutions from Graphs

This concept involves using the graph of a function to approximate inputs that produce a specific output. By locating the points where the function's graph attains a target value, one can estimate possible alternative solutions, which is especially useful when multiple inputs yield the same outcome.

Transcript

00:01 Alright, this problem is asking us to graph something and then interpret something on it.

00:08 So a company that produces calculators estimated that the profit p in dollars from selling a particular model of calculator was this.

00:16 P equals negative 152 x cubed plus 7545 x squared minus 169 ,625.

00:32 And they gave a restriction on that domain.

00:36 It's going to help us graph here.

00:38 X had to be between 0 and 45.

00:40 So they're saying x is the advertising expense in tens of thousands of dollars.

00:46 So keep that in mind.

00:47 So for this model of calculator, the advertising expenses was $400 ,000.

00:52 So x equals 40 and that profit was $2 ,174 ,375.

00:59 And they want to use a graphing utility to graph the profit function.

01:03 So i did that and i'm going to bring those over.

01:08 So on a ti calculator, you go into your y equals and i typed the p equation.

01:14 So this guy here is going to be my y sub 1.

01:17 And then my y sub 2, what i did was, no, for pete's sakes, why won't that graph right? there we go.

01:26 And for y sub 1, i have the original.

01:29 And for y sub 2, i have the other 2 ,000 ,000, what was it? 2 ,174 ,375.

01:37 That way i can do two things at once.

01:40 The other piece of the puzzle here is to get the window right.

01:44 They gave you the x values for the domain restrictions between 0 and 45.

01:48 And i just made this scale 5.

01:50 But then i also knew that the y intercept was down below negative 169 ,000.

01:56 So i made the minimum negative 200 ,000.

02:01 Then because that y sub 2 was over 2 ,000 ,000, i made the y max 3 ,000 ,000.

02:08 And then i just scaled it to be 100 ,000.

02:11 So what we are looking at here is that these points of intersection, right? so here's this is part a.

02:19 Part a tells you to graph it.

02:20 And this is now graphed.

02:22 Then it says use the graph from part a to estimate another amount the company could have spent on advertising that would have produced the same profit.

02:29 So remember right here, this guy here is where x was 40, right? and at 40, it made this profit of over 2 ,174 ,000, right? so what i did is i did second trace intersect, second calc intersect.

02:47 And i found this other intersection point over here...

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