Question 1

Paper 1Hard6 marks

During an acoustic test, sound level LL in decibels (dB), is defined by

L=10log⁡10(II0),L=10\log_{10}\left(\frac{I}{I_0}\right),

where II is the sound intensity and I0I_0 is a fixed reference intensity. When machines operate together, their sound intensities add.

Machine A alone produces a sound level of 67.467.4 dB and machine B alone produces a sound level of 72.172.1 dB.

(a)

Show that if the sound levels of A and B are LAL_A and LBL_B respectively, then

IBIA=10(LB−LA)/10.\frac{I_B}{I_A}=10^{(L_B-L_A)/10}.
[2]
(b)

Determine the sound level when A and B operate together. Give your answer to the nearest 0.10.1 dB.

[2]
(c)

Identical copies of machine A are used instead. Determine the least whole number of copies needed for their combined sound level to be at least the level found in part (b).

[2]

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