IB Maths AI HL AI 1.3 Exponents, logarithms and equations questions

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Question 1

Paper 1Hard6 marks

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

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

where $I$ is the sound intensity and $I_0$ is a fixed reference intensity. When machines operate together, their sound intensities add.

Machine A alone produces a sound level of $67.4$ dB and machine B alone produces a sound level of $72.1$ dB.

(a)

Show that if the sound levels of A and B are $L_A$ and $L_B$ respectively, then

\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 $0.1$ 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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