21M.1.SL.TZ1.11

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If a shark is spotted near to Brighton beach, a lifeguard will activate a siren to warn swimmers.

The sound intensity, $I$ , of the siren varies inversely with the square of the distance, $d$ , from the siren, where $d > 0$ .

It is known that at a distance of $1.5$ metres from the siren, the sound intensity is $4$ watts per square metre ( ${W m}^{- 2}$ ).

Show that $I = \frac{9}{d^{2}}$ .

[2]

Sketch the curve of $I$ on the axes below showing clearly the point $(1.5, 4)$ .

[2]

Whilst swimming, Scarlett can hear the siren only if the sound intensity at her location is greater than $1.5 \times 10^{- 6}$ ${W m}^{- 2}$ .

Find the values of $d$ where Scarlett cannot hear the siren.

[2]

Markscheme / solution

$I = \frac{k}{d^{2}}$ (M1)

$4 = \frac{k}{1 . 5^{2}}$ M1

$I = \frac{9}{d^{2}}$ AG

Note: The AG line must be seen for the second M1 to be awarded.
Award no marks for substituting $1.5$ and $4$ into $I = \frac{9}{d^{2}}$ (i.e., working backwards).

[2 marks]

A1A1

Note: Award A1 for correct general shape (concave up) with no $I$ -intercept, passing through the marked point $(1.5, 4)$ ; the point must be labelled with either the coordinates or the values $1.5$ and $4$ on the $x$ and $y$ axes. Award A1 for the curve showing asymptotic behavior (i.e. $I$ tends to $0$ , as $d$ tends to infinity), extending to at least $d = 6$ ; the curve must not cross nor veer away from the horizontal asymptote.

[2 marks]

$1.5 \times 10^{- 6} \geq \frac{9}{d^{2}}$ (M1)

Note: Award (M1) for a correct inequality.

$d \geq 2450 (m) (2449.48 \dots)$ A1

Note: Award A0 for $d = 2450$ .

[2 marks]

Examiners’ report

[N/A]