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Date	November 2019	Marks available	2	Reference code	19N.2.SL.TZ0.T_6
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	T_6	Adapted from	N/A

Question

The Maxwell Ohm Company is designing a portable Bluetooth speaker. The speaker is in the shape of a cylinder with a hemisphere at each end of the cylinder.

The dimensions of the speaker, in centimetres, are illustrated in the following diagram where $r$ is the radius of the hemisphere, and $l$ is the length of the cylinder, with $r > 0$ and $l \geq 0$ .

The Maxwell Ohm Company has decided that the speaker will have a surface area of $300 {cm}^{2}$ .

The quality of sound from the speaker will improve as $V$ increases.

Write down an expression for $V$ , the volume (cm³) of the speaker, in terms of $r$ , $l$ and $π$ .

[2]

Write down an equation for the surface area of the speaker in terms of $r$ , $l$ and $π$ .

[3]

Given the design constraint that $l = \frac{150 - 2 π r^{2}}{π r}$ , show that $V = 150 r - \frac{2 π r^{3}}{3}$ .

[2]

Find $\frac{d V}{d r}$ .

[2]

Using your answer to part (d), show that $V$ is a maximum when $r$ is equal to $\sqrt{\frac{75}{π}} cm$ .

[2]

Find the length of the cylinder for which $V$ is a maximum.

[2]

Calculate the maximum value of $V$ .

[2]

Use your answer to part (f) to identify the shape of the speaker with the best quality of sound.

[1]

Markscheme

$(V =) \frac{4 π r^{3}}{3} + π r^{2} l$ (or equivalent) (A1)(A1)

Note: Award (A1) for either the volume of a hemisphere formula multiplied by $2$ or the volume of a cylinder formula, and (A1) for completely correct expression. Accept equivalent expressions. Award at most (A1)(A0) if $h$ is used instead of $l$ .

[2 marks]

$300 = 4 π r^{2} + 2 π r l$ (A1)(A1)(A1)

Note: Award (A1) for the surface area of a hemisphere multiplied by $2$ . Award (A1) for the surface area of a cylinder. Award (A1) for the addition of their formulas equated to $300$ . Award at most (A1)(A1)(A0) if $h$ is used instead of $l$ , unless already penalized in part (a).

[3 marks]

$V = \frac{4 π r^{3}}{3} + π r^{2} (\frac{150 - 2 π r^{2}}{π r})$ (M1)

Note: Award (M1) for their correctly substituted formula for $V$ .

$V = \frac{4 π r^{3}}{3} + 150 r - 2 π r^{3}$ (M1)

Note: Award (M1) for correct expansion of brackets and simplification of the cylinder expression in $V$ leading to the final answer.

$V = 150 r - \frac{2 π r^{3}}{3}$ (AG)

Note: The final line must be seen, with no incorrect working, for the second (M1) to be awarded.

[2 marks]

$(\frac{d V}{d r} =) 150 - 2 π r^{2}$ (A1)(A1)

Note: Award (A1) for $150$ . Award (A1) for $- 2 π r^{2}$ . Award maximum (A1)(A0) if extra terms seen.

[2 marks]

$150 - 2 π r^{2} = 0$ OR $\frac{d V}{d r} = 0$ OR sketch of $\frac{d V}{d r}$ with $x$ -intercept indicated (M1)

Note: Award (M1) for equating their derivative to zero or a sketch of their derivative with $x$ -intercept indicated.

$r = \sqrt{\frac{150}{2 π}}$ OR $r^{2} = \frac{150}{2 π}$ (A1)

$r = \sqrt{\frac{75}{π}}$ (AG)

Note: The (AG) line must be seen for the preceding (A1) to be awarded.

[2 marks]

$(l =) \frac{150 - 2 π {(\sqrt{\frac{75}{π}})}^{2}}{π (\sqrt{\frac{75}{π}})}$ (M1)

Note: Award (M1) for correct substitution in the given formula for the length of the cylinder.

$(l =) 0 (cm)$ (A1)(G2)

Note: Award (M1)(A1) for correct substitution of the $3$ sf approximation $4.89$ leading to a correct answer of zero.

[2 marks]

$V = 150 (\sqrt{\frac{75}{π}}) - \frac{2 π {(\sqrt{\frac{75}{π}})}^{3}}{3}$ OR $V = \frac{4 π {(\sqrt{\frac{75}{π}})}^{3}}{3}$ (M1)

Note: Award (M1) for correct substitution in the formula for the volume of the speaker or the volume of a sphere.

$489 (488.602 \dots, 100 \sqrt{\frac{75}{π}}) ({cm}^{3})$ (A1)(G2)

Note: Accept $489.795 \dots$ from use of $3$ sf value of $\sqrt{\frac{75}{π}}$ . Award (M1)(A1)(ft) for correct substitution in their volume of speaker. Follow through from parts (a) and (f).

[2 marks]

sphere (spherical) (A1)(ft)

Note: Question requires the use of part (f) so if there is no answer to part (f), part (h) is awarded (A0). Follow through from their $l > 0$ .

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.1—3d space, volume, angles, midpoints

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