5.3 Modeling with polynomial functions

Introduction

Volume, surface area, and constraints

Other applications of polynomial functions

A piece of cardboard that is $12 \times 18$ (each measured in inches) is being made into a box without a top. To do so, squares are cut from each corner of the cardboard and the remaining sides are folded up.

Let $x$ be the side length of the squares being cut from the corners of the cardboard. Draw a labeled diagram that shows the given information and the variable being used.

Label the provided picture, using $x$ for the length of each side of the square end, and $y$ for the other edge of the package.

How does the length plus girth of $120$ inches result in an equation (often called a constraint equation) that relates $x$ and $y$? Explain, and state the equation.

Solve the equation you found in (b) for one of the variables present.

Hence determine the volume, $V$, of the package as a function of a single variable.

What is the domain of the function $V$ in the context of the physical setting of this problem? (Hint: neither $x$ nor $y$ can equal $0$.)

Use the formula for the surface area of a cylinder and the given constraint that the can's surface area is $60$ square inches to write an equation that connects the radius $r$ and height $h$.

Solve the equation you found in (a) for $h$ in terms of $r$.

Recall that the volume of a cylinder is $V = \pi r^2 h$. Use your work in (b) to write $V$ as a function of the single variable $r$; simplify the formula as much as possible.

What is the domain of the function $V$ in the context of the physical setting of this problem? (Hint: how does the constraint on surface area provide an upper bound for the value of $r$? Think about the maximum area that can be allocated to the top and bottom of the can.)

For what values of $x$ does it appear that $\sin(x) \approx T_1(x)$?

For what values of $x$ does it appear that $\sin(x) \approx T_3(x)$?

For what values of $x$ does it appear that $\sin(x) \approx T_5(x)$?

What overall trend do you observe? How good is the approximation generated by $T_{19}(x)$?

In a new Desmos worksheet, plot the function $y = \cos(x)$ along with the following functions: $P_2(x) = 1 - \frac{x^2}{2!}$ and $P_4(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}$. Based on the patterns with the coefficients in the polynomials approximating $\sin(x)$ and the polynomials $P_2$ and $P_4$ here, conjecture formulas for $P_6$, $P_8$, and $P_{18}$ and plot them. How well can we approximate $y = \cos(x)$ using polynomials?