Forming Equations: WJEC GCSE Higher June 2022 Unit 2 Q8

Higher June 2022 Unit 2 Q8

WJECHigherCurrent spec6 marksForming Equations Iterations Volume & Surface Area

8. This cuboid has:

length = 5 cm
width = \(x\) cm
height = \((x^2 + 3)\) cm
volume = 132 cm³.

Cuboid with length 5, width x and height x squared plus 3

Diagram not drawn to scale

(a) Show that \(5x^3 + 15x = 132\). [1]

(b)

(i) A solution of the equation \[5x^3 + 15x = 132\] lies between 2 and 3.
Use the method of trial and improvement to find this solution correct to 1 decimal place.
You must show all your working. [4]

(ii) Hence, find the height of the cuboid. [1]

Height of the cuboid = …………… cm

Mark scheme (a)

Scheme	Marks	Comments
Any intention of length × width × height = 132 e.g. \(5x\,(x^2 + 3) = 132\) \(5 \times x \times (x^2 + 3) = 132\) or \(5x \times (x^2 + 3) = 132\) or equivalent	B1	Must be = 132. May be seen in parts. Do not allow missing brackets e.g. \(5 \times x \times x^2 + 3 = 132\).

Mark scheme (b)

(b)(i)

Scheme	Marks	Comments
One correct evaluation \(2 \leqslant x \leqslant 3\)	B1
2 correct evaluations \(2{\cdot}55 \leqslant x \leqslant 2{\cdot}75\), (one value < 132, one value > 132)	B1
2 correct evaluations \(2{\cdot}55 \leqslant x \leqslant 2{\cdot}65\), (one value < 132, one value > 132)	M1
\(x = 2{\cdot}6\)	A1

Comments

Correct evaluation regarded as enough to identify if <132 or >132. If evaluations not seen accept ‘too high’ or ‘too low’.
Look out for testing \(5x^3 + 15x - 132 = 0\) or \(x^3 + 3x = 26{\cdot}4\) or equivalent

\(x\)	\(5x^3 + 15x\)
2	70
2.1	77.805
2.2	86.24
2.3	95.335
2.4	105.12
2.5	115.625	2.55	121.1568…
2.6	126.88	2.65	132.798…..
2.7	138.915	2.75	145.234…...
2.8	151.76
2.9	165.445
3	180

The rows 2.6 and 2.7 and the entry 2.65, 132.798….. were shaded and in bold in the original.

(b)(ii)

Scheme	Marks	Comments
An answer in the range 9·76 to 10·16 (cm)	B1	Answer may be shown on the diagram. FT ‘their 2·6’² + 3. FT 132 ÷ (5 × ‘their x’).