17M.2.SL.TZ2.T_5

pestleMathematicsAASLPaper 217M· sl-5-8-testing-for-max-and-min-optimisation-points-of-inflexionsource ↗

Violeta plans to grow flowers in a rectangular plot. She places a fence to mark out the perimeter of the plot and uses 200 metres of fence. The length of the plot is $x$ metres.

M17/5/MATSD/SP2/ENG/TZ2/05

Violeta places the fence so that the area of the plot is maximized.

By selling her flowers, Violeta earns 2 Bulgarian Levs (BGN) per square metre of the plot.

Show that the width of the plot, in metres, is given by $100 - x$ .

[1]

Write down the area of the plot in terms of $x$ .

[1]

Find the value of $x$ that maximizes the area of the plot.

[2]

Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.

[2]

Markscheme / solution

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

$\frac{{200 - 2x}}{2}$ (or equivalent) (M1)

$2x + 2y = 200$ (or equivalent) (M1)

Note: Award (M1) for a correct expression leading to $100 - x$ (the $100 - x$ does not need to be seen). The 200 must be seen for the (M1) to be awarded. Do not accept $100 - x$ substituted in the perimeter of the rectangle formula.

$100 - x$ (AG)

[1 mark]

$({\text{area}} = ){\text{ }}x(100 - x)$ $\,\,\,$ OR $\,\,\,$ $- {x^2} + 100x$ (or equivalent) (A1)

[1 mark]

$x = \frac{{ - 100}}{{ - 2}}$ $\,\,\,$ OR $\,\,\,$ $- 2x + 100 = 0$ $\,\,\,$ OR $\,\,\,$ graphical method (M1)

Note: Award (M1) for use of axis of symmetry formula or first derivative equated to zero or a sketch graph.

$x = 50$ (A1)(ft)(G2)

Note: Follow through from part (b), provided x is positive and less than 100.

[2 marks]

$50(100 - 50) \times 2$ (M1)(M1)

Note: Award (M1) for substituting their $x$ into their formula for area (accept “ $50 \times 50$ ” for the substituted formula), and (M1) for multiplying by 2. Award at most (M0)(M1) if their calculation does not lead to 5000 (BGN), although the 5000 (BGN) does not need to be seen explicitly.

Substitution of 50 into area formula may be seen in part (c).

5000 (BGN) (AG)

[2 marks]

Examiners’ report

[N/A]