17M.1.SL.TZ2.T_12

pestleMathematicsAASLPaper 117M· sl-3-1-3d-space-volume-angles-distance-midpointssource ↗

A cylindrical container with a radius of 8 cm is placed on a flat surface. The container is filled with water to a height of 12 cm, as shown in the following diagram.

M17/5/MATSD/SP1/ENG/TZ2/12

A heavy ball with a radius of 2.9 cm is dropped into the container. As a result, the height of the water increases to $h$ cm, as shown in the following diagram.

M17/5/MATSD/SP1/ENG/TZ2/12.b

Find the volume of water in the container.

[2]

Find the value of $h$ .

[4]

Markscheme / solution

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

$\pi \times {8^2} \times 12$ (M1)

Note: Award (M1) for correct substitution into the volume of a cylinder formula.

$2410{\text{ c}}{{\text{m}}^3}{\text{ }}(2412.74 \ldots {\text{ c}}{{\text{m}}^3},{\text{ }}768\pi {\text{ c}}{{\text{m}}^3})$ (A1) (C2)

[2 marks]

$\frac{4}{3}\pi \times {2.9^3} + 768\pi = \pi \times {8^2}h$ (M1)(M1)(M1)

Note: Award (M1) for correct substitution into the volume of a sphere formula (this may be implied by seeing 102.160…), (M1) for adding their volume of the ball to their part (a), (M1) for equating a volume to the volume of a cylinder with a height of $h$ .

$\frac{4}{3}\pi \times {2.9^3} = \pi \times {8^2}(h - 12)$ (M1)(M1)(M1)

Note: Award (M1) for correct substitution into the volume of a sphere formula (this may be implied by seeing 102.160…), (M1) for equating to the volume of a cylinder, (M1) for the height of the water level increase, $h - 12$ . Accept $h$ for $h - 12$ if adding 12 is implied by their answer.

$(h = ){\text{ }}12.5{\text{ (cm) }}\left( {12.5081 \ldots {\text{ (cm)}}} \right)$ (A1)(ft) (C4)

Note: If 3 sf answer used, answer is 12.5 (12.4944…). Follow through from part (a) if first method is used.

[4 marks]

Examiners’ report

[N/A]