Question 1

Paper 2Medium13 marks

At a rehearsal venue, the intensity II in W m−2\text{W m}^{-2} of a short electronic tone tt seconds after it ends is modelled by

I(t)=I0e−kt,t≥0,I(t)=I_0e^{-kt},\qquad t\geq 0,

where I0=8.00×10−5I_0=8.00\times10^{-5} A sensor reading gives I(5)=2.40×10−5I(5)=2.40\times10^{-5}

The sound level LL in decibels (dB), is modelled by

L=10log⁡10(IIref),Iref=1.00×10−12 W m−2.L=10\log_{10}\left(\frac{I}{I_{\mathrm{ref}}}\right), \qquad I_{\mathrm{ref}}=1.00\times10^{-12}\text{ W m}^{-2}.

The venue’s policy is to begin recording only when the predicted sound level is at most 70.070.0 dB.

(a)

For the initial intensity:

(i) Write I0Iref\dfrac{I_0}{I_{\mathrm{ref}}} in standard form.

(ii) Calculate L(0)L(0) giving your answer to three significant figures.

[3]
(b)

Determine kk giving your answer to three significant figures and including its unit.

[3]
(c)

Show that

L(t)=L(0)−10ktlog⁡10(e).L(t)=L(0)-10kt\log_{10}(e).
[2]
(d)

Hence or otherwise, determine the first time at which the predicted sound level is at most 70.070.0 dB. Give your answer to the nearest 0.10.1 s.

[3]
(e)

A manager claims that recording can begin 8.08.0 seconds after the tone ends. Assess this claim using the model, and state one assumption or limitation relevant to the decision.

[2]

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