Volume & Surface Area: WJEC GCSE Higher November 2021 Unit 1 Q15

Higher November 2021 Unit 1 Q15

WJECHigherCurrent spec5 marksVolume & Surface Area

15. A solid object is made out of a hemisphere and a cylinder.

Diagram not drawn to scale

The radius of the common circular surface is 3 cm.
The volume of the whole object is \(63\pi\) cm³.

Calculate the total height of the object. [5]

From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\)

Mark scheme

Scheme	Marks	Comments
\(\pi \times 3^2 \times h \;+\; \tfrac{2}{3} \times \pi \times 3^3 = 63\pi\) or equivalent Allow \(\dfrac{\frac{4}{3} \times \pi \times 3^3}{2}\) for \(\tfrac{2}{3} \times \pi \times 3^3\).	M2	(Using ‘h’ as height of cylinder) M1 for summing 2 terms and equating to \(63\pi\), with 1 term being correct M1 may be implied by a subtraction or seen in stages e.g. \(9\pi h = 27\pi\) from \(63\pi - 36\pi\) or \(9\pi h = 36\pi\) from \(63\pi - 27\pi\) or \(9\pi h = 57\pi\) from \(63\pi - 6\pi\) (using incorrect evaluations for volume of hemisphere) Allow the use of \(\pi = 3{\cdot}14\).
\(9\pi h = 63\pi - 18\pi\) or \(h = \dfrac{63\pi - 18\pi}{9\pi}\) or equivalent e.g. \(\dfrac{45\pi}{9\pi}\)	m1	Isolating the term in h. FT from M1 or M2.
= 5 (cm)	A1	C.A.O.
(Total height =) 8 (cm)	B1	FT ‘their 5’ + 3 provided M1m1 or M2m1 awarded If no marks, award SC1 for \(18\pi\) for the volume of the hemisphere (but NOT from a calculation for surface area) OR SC2 for \(45\pi\) for the volume of the cylinder.

Scheme	Marks	Comments
\(\pi \times 3^2 \times (H - 3) \;+\; \tfrac{2}{3} \times \pi \times 3^3 = 63\pi\) or equivalent Allow \(\dfrac{\frac{4}{3} \times \pi \times 3^3}{2}\) for \(\tfrac{2}{3} \times \pi \times 3^3\).	M2	(Using ‘H’ as total height of object) M1 for summing 2 terms and equating to \(63\pi\), with 1 term being correct. M1 may be implied by a subtraction or seen in stages. Allow the use of \(\pi = 3{\cdot}14\).
\(9\pi(H - 3) = 63\pi - 18\pi\) or equivalent e.g \(9\pi H = 63\pi - 18\pi + 27\pi\)	m1	Isolating the term in (H – 3). FT from M1 or M2.
\((H =)\ \dfrac{63\pi - 18\pi + 27\pi}{9\pi}\) or equivalent e.g. \(\dfrac{72\pi}{9\pi}\)	A1	FT from M1m1 or M2m1.
(Height of object =) 8 (cm)	A1	C.A.O. If no marks, award SC1 for \(18\pi\) for the volume of the hemisphere (but NOT from a calculation for surface area) OR SC2 for \(45\pi\) for the volume of the cylinder OR SC2 for an appropriate volume of \(72\pi\).