Modeling with Polynomials

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Worksheet Modeling with Polynomials

1.

For the following exercise, use the written statement to construct a polynomial function that represents the required information.

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An oil slick is expanding as a circle. The radius of the circle is currently 3.5 inches and is increasing at a rate of 10 inches per hour. Express the area of the circle, $A$ as a function of $h\text{,}$ the number of hours elapsed. ( Answer should be $A(h) =$ some function of $h\text{,}$ enter $\pi$ as pi )

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The polynomial is:

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Answer.

$A\mathopen{}\left(h\right) = \pi \mathopen{}\left(10h+3.5\right)^{2}$

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2.

A rectangle has a length of 18 units and a width of 5 units. Squares of `x` by `x` units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of `x`.

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$f(x) =$

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Answer.

$x\mathopen{}\left(18-2x\right)\mathopen{}\left(5-2x\right)$

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3.

A right circular cone has a radius of ${5x+3}$ and a height `2` units less than its radius. Express the volume of the cone as a polynomial function. The volume of a cone is `V = 1/3 pi r^2 h` for radius $r$ and height $h\text{.}$ Enter $\pi$ as pi.

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$V(x) =$

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Answer.

$\frac{1}{3}\pi \mathopen{}\left(5x+3\right)^{2}\mathopen{}\left(5x+1\right)$

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4.

A market analyst finds that if a company produces and sells $x$ blenders annually, the total profit in dollars is

\begin{equation*} P(x) = 10 x+ 0.3 x^2 - 0.0013 x^3 - 355 \end{equation*}

Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following.

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When just a few blenders are produced, the company loses money (i.e., profit is negative). For example $P(10) = -226.3\text{,}$ so the company loses $226.30 if it produces and sells only 10 blenders. How many blenders must the company produce to break even?

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Number of blenders =

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Does the profit increase indefinitely as more blenders are produced and sold, or is there a largest possible profit the firm could earn? If there is a maximum profit, enter that value. If profit could increase indefinitely, enter None.

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Maximum profit = $

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Answer 1.

$23$

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Answer 2.

$3628.27420096111$

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5.

An auto company’s sales volume can be modeled by ${4.6x^{2}+8.2x+1.7}\text{,}$ and its cost can be modeled by ${2.2x^{2}+1.8x+1.7}\text{,}$ where $x$ represents the number of cars produced, and $y$ stands for money in thousand dollars. We can calculate the company’s net profit by subtracting cost from sales. Find the polynomial which models the company’s sales in thousands of dollars.

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The company’s profit can be modeled by dollars.

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Answer.

$2.4x^{2}+6.4x$

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Solution.

We will subtract the cost polynomial from the sales polynomial:

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$\displaystyle{ \begin{aligned} \amp \phantom{{}=}\left({4.6x^{2}+8.2x+1.7}\right)-\left({2.2x^{2}+1.8x+1.7}\right) \\ \amp = \left(4.6x^2-2.2x^2\right)+\left({8.2}x-1.8 x \right)+\left(1.7-1.7 \right)\\ \amp = {2.4x^{2}+6.4x} \end{aligned} }$

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The company’s profit can be modeled by ${2.4x^{2}+6.4x}$ dollars.

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6.

A farmer is building fence around a triangular area. The cost of building the shortest side is ${30x}$ dollars, where $x$ stands for the length of the side in feet. The cost of building the other two sides can be modeled by ${9x^{2}-4x+40}$ dollars and ${2x^{3}-1.5x+35}$ dollars, respectively. What’s the total cost of building fence for all three sides?

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The cost of building fence for all three sides would be dollars.

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Answer.

$2x^{3}+9x^{2}+24.5x+75$

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Solution.

We will subtract the cost polynomial from the sales polynomial:

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$\displaystyle{ \begin{aligned} \amp \phantom{{}=}\left({30x}\right)+\left({9x^{2}-4x+40}\right)+\left({2x^{3}-1.5x+35}\right) \\ \amp = 2x^3+9x^2+\left(30x+(-4 x)+(-1.5 x) \right)+\left(40+35 \right)\\ \amp = {2x^{3}+9x^{2}+24.5x+75} \end{aligned} }$

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The cost of building fence for all three sides would be ${2x^{3}+9x^{2}+24.5x+75}$ dollars.

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7.

An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial ${3x^{4}+13x^{2}+0.5x}\text{,}$ and the area of the house’s base can be modeled by ${2x^{3}+0.5x+40}\text{.}$ The rest of the plot is the yard. What’s the yard’s area?

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The area of the yard can be modeled by the polynomial .

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Answer.

$3x^{4}-2x^{3}+13x^{2}-40$

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Solution.

To find the yard’s area, we subtract the house base’s area from the plot’s area, and we have:

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$\displaystyle{ \begin{aligned} \amp \phantom{{}=}\left({3x^{4}+13x^{2}+0.5x}\right)-\left({2x^{3}+0.5x+40}\right) \\ \amp = 3x^4 - 2x^3+13x^2+\left(0.5x-0.5 x \right)+\left(0-40 \right)\\ \amp = {3x^{4}-2x^{3}+13x^{2}-40} \end{aligned} }$

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The area of the yard can be modeled by the polynomial ${3x^{4}-2x^{3}+13x^{2}-40}\text{.}$

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8.

Answer 1.

$-3x^{2}+48x$

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Answer 2.

$0 < x < 16$

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Answer 3.

$8$

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Answer 4.

$24$

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9.

An open box is to be made from a flat piece of material 14 inches long and 6 inches wide by cutting equal squares of length $x$from the corners and folding up the sides.

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Write the volume $V$of the box as a function of $x\text{.}$ Leave it as a product of factors, do not multiply out the factors.

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$V =$

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If we write the domain of the box as an open interval in the form (a,b), then what is $a =\text{?}$

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$a =$

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and what is $b =\text{?}$

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$b =$