Maximizing Volume with Polynomials

This lesson focuses on using polynomials to maximize the volume of a box under specific constraints. It begins with an introduction to cubic functions and the problem scenario, followed by defining variables for the box's dimensions. The lesson then simplifies expressions for these dimensions and creates a polynomial function. After graphing the function, the lesson analyzes the results to determine the dimensions that maximize volume. The lesson concludes with a summary and practice examples.

What is the primary objective of this lesson?

• To understand linear functions

• To maximize volume using polynomials

• To solve algebraic equations

• To learn about quadratic equations

What type of function is used to represent volume in this lesson?

• Quadratic function

• Linear function

• Cubic function

• Exponential function

In the example problem, what is the maximum sum of the dimensions of the box?

• 10 inches

• 12 inches

• 8 inches

• 6 inches

What variable is used to represent the height of the box?

• x

• z

• h

• y

How is the length of the box related to its height?

• It is 1.5 times the height

• It is half the height

• It is twice the height

• It is equal to the height

What is the expression for the width of the box?

• 6 - 2.5x

• 6 - 3x

• 6 - 1.5x

• 6 - x

What is the first step in simplifying the polynomial function for volume?

• Divide by x

• Add the coefficients

• Multiply x by 1.5x

• Subtract 1.5x from 6

What is the coefficient of the x^3 term in the polynomial function?

• 6

• 1.5

• 2.5

• 3.75

What tool is used to graph the polynomial function?

• Desmos

• Excel

• Matlab

• Graphing calculator

What is the maximum volume achieved according to the graph?

• 9.68

• 8.68

• 7.68

• 6.68