19M.1.SL.TZ2.T_15

pestleMathematicsAASLPaper 119M· sl-5-3-differentiating-polynomials-n-e-zsource ↗

A potter sells $x$ vases per month.

His monthly profit in Australian dollars (AUD) can be modelled by

$P\left( x \right) = - \frac{1}{5}{x^3} + 7{x^2} - 120{\text{,}}\,\,x \geqslant 0.$

Find the value of $P$ if no vases are sold.

[1]

Differentiate $P\left( x \right)$ .

[2]

Hence, find the number of vases that will maximize the profit.

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

−120 (AUD) (A1) (C1)

[1 mark]

$- \frac{3}{5}{x^2} + 14x$ (A1)(A1) (C2)

Note: Award (A1) for each correct term. Award at most (A1)(A0) for extra terms seen.

[2 marks]

$- \frac{3}{5}{x^2} + 14x = 0$ (M1)

Note: Award (M1) for equating their derivative to zero.

sketch of their derivative (approximately correct shape) with $x$ -intercept seen (M1)

$23\frac{1}{3}\,\,\,\left( {23.3{\text{,}}\,\,23.3333 \ldots {\text{,}}\,\,\frac{{70}}{3}} \right)$ (A1)(ft)

Note: Award (C2) for $23\frac{1}{3}\,\,\,\left( {23.3{\text{,}}\,\,23.3333 \ldots {\text{,}}\,\,\frac{{70}}{3}} \right)$ seen without working.