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 compost bay. Let \(p\) be its height, measured in m

The base length is \(-2p + 18\), the base width is \(-p + 10\), and the height is \(p\). All three lengths use m

The volume is the product of these three lengths

P(p)=\left(p\right)\left(-2p + 18\right)\left(-p + 10\right)

The stated operating conditions give the domain \(0 \le p \le 9\)

All displayed quantities are exact

\(P(p)\) denotes the volume, measured in \(\text{m}^3\)

The stated closed domain includes zero-volume boundary cases of the mathematical model. A non-degenerate rectangular prism requires all three lengths to 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 \(p\) in base length Constant in base width coefficient of \(p\) in base width

(b)1 mark

Select the cubic product that follows from the stated component rules

P(p)=\left(p\right)\left(2p + 18\right)\left(-p + 10\right)

P(p)=\left(p\right)\left(-2p + 18\right)

P(p)=\left(-2p + 28\right)

P(p)=\left(p\right)\left(-2p + 18\right)\left(-p + 10\right)

State the exact lower and upper bounds of the domain

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

(d)1 mark

Find the exact volume for \(p=9/2\)

\(P(\frac{9}{2})\) (m^3)

Format: Exact value

(e)1 mark

Should the proposed input \(p=12\) be used in this model? Select the statement with the correct reason

Use \(p=12\) because the rule gives an output, even though the original input restriction is not checked

Do not use \(p=12\) because it lies outside the stated domain \(0 \le p \le 9\)

Use \(p=12\) because it lies in the stated domain \(0 \le p \le 9\)

Use \(p=12\) because a cubic rule is defined for every real input