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IB Mathematics SL 1.1 Operations with numbers AI HL Paper 1- Exam Style Questions- New Syllabus

IBDP Maths AI HL Paper 1 - All Topics

Question

The perceived loudness of a sound, \(L\), measured in decibels (dB), is related to its physical intensity, \(I\), measured in watts per square metre (\(Wm^{-2}\)).
This relationship is modelled by the logarithmic function:
\(L = 10 \log_{10}\left(\frac{I}{I_0}\right), I > 0\)
where \(I_0\) represents the threshold of human hearing (the reference intensity).
The reference intensity \(I_0\) is given as \(10^{-12} \, Wm^{-2}\).
An acoustic engineer measures the intensity of sound on a crowded urban street to be \(10^{-5} \, Wm^{-2}\).
(a) Determine the loudness of the sound on this street.
At a nearby airport, a jet engine is found to produce a loudness of 185 dB.
(b) Calculate the intensity of the jet engine’s sound. Express your answer in scientific notation \(a \times 10^k\), where \(1 \le a < 10\) and \(k \in \mathbb{Z}\).

Most-appropriate topic codes:

• SL 1.5: Logarithms with base 10 and numerical evaluation — part (a)
• SL 1.1: Operations with numbers in the form \(a \times 10^k\) — part (b)
▶️ Answer/Explanation
Detailed solution

(a)
Substitute the given values \(I = 10^{-5}\) and \(I_0 = 10^{-12}\) into the formula:
\(L = 10 \log_{10}\left(\frac{10^{-5}}{10^{-12}}\right)\)
Using exponent laws: \(\frac{10^{-5}}{10^{-12}} = 10^{-5 – (-12)} = 10^7\).
\(L = 10 \log_{10}(10^7)\)
Using log laws (\(\log_{10}(10^k) = k\)):
\(L = 10 \times 7 = 70\).
Loudness = 70 dB.

(b)
Substitute \(L = 185\) into the formula:
\(185 = 10 \log_{10}\left(\frac{I}{10^{-12}}\right)\)
Divide by 10:
\(18.5 = \log_{10}\left(\frac{I}{10^{-12}}\right)\)
Convert from log form to exponential form (\(y = \log_b x \iff x = b^y\)):
\(\frac{I}{10^{-12}} = 10^{18.5}\)
Solve for \(I\):
\(I = 10^{18.5} \times 10^{-12}\)
\(I = 10^{6.5}\)
Calculate value:
\(I \approx 3,162,277.66\)
Convert to scientific notation \(a \times 10^k\):
\(I = 3.16 \times 10^6 \, Wm^{-2}\).

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