1 of 8 0-day streak

Formulate and interpret a cubic model

Question 1 of 8

Calculator allowed

Consider a rectangular-prism model for the rectangular panel. Let \(w\) be its height, measured in cm

The base length is \(-w + 7\), the base width is \(-2w + 18\), and the height is \(w\). All three lengths use cm

The volume is the product of these three lengths

P(w)=\left(w\right)\left(-w + 7\right)\left(-2w + 18\right)

The stated operating conditions give the domain \(0 < w < 7\)

All displayed quantities are exact

\(P(w)\) denotes the volume, measured in \(\text{cm}^3\)

The open domain excludes zero-volume boundary cases: all three lengths must be positive

(a)4 marks

Record the exact constants and coefficients in the two component rules before expanding the product

Constant in base length coefficient of \(w\) in base length Constant in base width coefficient of \(w\) in base width

(b)1 mark

Identify the exact product rule before evaluating the model

P(w)=\left(-2w + 25\right)

P(w)=\left(w\right)\left(-w + 7\right)\left(-2w + 18\right)

P(w)=\left(w\right)\left(w + 7\right)\left(-2w + 18\right)

P(w)=\left(w\right)\left(-w + 7\right)

Give the exact lower and upper bounds for the independent quantity

lower endpoint of the \(w\)-domain upper endpoint of the \(w\)-domain

(d)1 mark

Substitute the stated input and record the exact model value

\(P(\frac{7}{2})\) (cm^3)

Format: Exact value

(e)1 mark

Should the proposed input \(w=28/3\) be used in this model? Select the statement with the correct reason

Use \(w=\frac{28}{3}\) because the rule gives an output, even though the original input restriction is not checked

Use \(w=\frac{28}{3}\) because a cubic rule is defined for every real input

Use \(w=\frac{28}{3}\) because it lies in the stated domain \(0 < w < 7\)

Do not use \(w=\frac{28}{3}\) because it lies outside the stated domain \(0 < w < 7\)