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Formulate and interpret a cubic model

Start a set of 8
Step 5 of 5

For a tray cut from a 10 by 8 unit sheet, including zero-volume boundary cases, which cut-width domain is valid?

Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, Calculator allowedBuild a worksheet

Building a cubic model

Express each component in one independent variable, multiply the factors and derive the domain from the original conditions

\[V(x)=x\left(10-2x\right)\left(8-2x\right),\qquad 0\le x\le4\]
  1. Define the input Choose the independent quantity and state what the model output represents \(\displaystyle x=\text{cut width},\quad V=V(x)\)
  2. Write components in x Express every changing length or factor using the same independent quantity \(\displaystyle 10-2x,\quad 8-2x\)
  3. Multiply complete factors Use all component factors in the product before expanding or evaluating \(\displaystyle V(x)=x\left(10-2x\right)\left(8-2x\right)\)
  4. Keep the valid domain Use the closed mathematical domain when zero-volume boundary cases are included. A non-degenerate tray requires 0<x<4 \(\displaystyle 0\le x\le4\)

Worked example

A 12 cm by 9 cm rectangular sheet is used to form an open tray by cutting out corner squares and folding the sides. For cut width x cm, V(x)=x(12-2x)(9-2x). Include zero-volume boundary cases in the mathematical domain. Find that domain and V(2), and decide whether x=5 is valid

\[V(x)=x\left(12-2x\right)\left(9-2x\right)\]
  1. \(\displaystyle x\ge0,\quad 12-2x\ge0,\quad 9-2x\ge0\)The closed mathematical model permits non-negative lengths. A non-degenerate tray requires all three lengths to be positive
  2. \(\displaystyle 0\le x\le\frac{9}{2}\)The shorter original dimension gives the strongest upper bound
  3. \(\displaystyle V(2)=2\times8\times5=80\ \text{cm}^3\)At x=2, the three factors are 2 cm, 8 cm and 5 cm, giving 80 cubic centimetres
  4. \(\displaystyle 5>\frac{9}{2}\quad\Rightarrow\quad x=5\text{ is outside the model domain}\)The value x=5 is not valid because it exceeds the upper bound set by the 9 cm side

Common mistakes

Ignoring the model domain
✕\(\displaystyle x\in\mathbb{R}\)✓\(\displaystyle 0\le x\le4\)
For this mathematical model, all three lengths are non-negative and zero-volume boundary cases are included, giving 0≤x≤4. A non-degenerate tray requires 0<x<4
Keeping an invalid input
✕\(\displaystyle V(5)=5\times0\times(-2)=0\quad\Rightarrow\quad x=5\text{ is valid}\)✓\(\displaystyle 5>4\quad\Rightarrow\quad x=5\text{ is outside }[0,4]\)
An algebraic output does not make an input valid when it violates the original length conditions
Adding the factors
✕\(\displaystyle V(2)=2+6+4=12\)✓\(\displaystyle V(2)=2\times6\times4=48\)
The model is a product of the three component factors, so the factors must be multiplied

Difficulty levels

introUse component relationships to write a cubic product and state its basic domain
standardEvaluate a product model or empirical cubic at a valid input
challengeUse a cubic model with thresholds or several conditions to restrict valid inputs
examComplete the modelling chain from relationships to domain, evaluation and interpretation

Questions about this topic

Why is the domain part of the model?+

The domain records the conditions under which the component relationships are meaningful. A cubic expression outside that domain is only an algebraic extension

Why are the component factors multiplied?+

The model represents a product of dimensions or other linked quantities. Each factor contributes to the output, so adding them changes the model

Can a model be used outside its stated domain?+

Only as an explicitly qualified extrapolation. Conclusions about the original situation must use inputs that satisfy its physical or operating conditions

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 4: Cubic polynomials

In this chapter

Recognise and classify a polynomialEvaluate a polynomial or recover a coefficient parameterAdd, subtract or multiply polynomialsDivide polynomials and state quotient and remainderAnalyse a cubic graph or determine its ruleDetermine parameters in a polynomial identityApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsSolve a cubic inequalityIntersections of a cubic with a line or quadraticFormulate and interpret a cubic model