IB Mathematics SL 2.3 The graph of a function AI SL Paper 1- Exam Style Questions- New Syllabus
Question
If a shark is spotted near Sunny Coast 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\(^{-2}\)).
(a) Demonstrate that \( I = \dfrac{9}{d^2} \). [2]
(b) Sketch the curve of \( I \), showing clearly the point (1.5, 4). [2]
(c) Whilst swimming, Jordan can hear the siren only if the sound intensity at their location is greater than \( 1.5 \times 10^{-6} \) W m\(^{-2}\). Determine the values of \( d \) where Jordan cannot hear the siren. [2]
▶️ Answer/Explanation
Markscheme
(a)
Sound intensity varies inversely with the square of distance: \( I = \dfrac{k}{d^2} \). M1
Given: \( I = 4 \) when \( d = 1.5 \). Substitute: \( 4 = \dfrac{k}{1.5^2} \).
Calculate: \( 1.5^2 = 2.25 \), so \( 4 = \dfrac{k}{2.25} \), \( k = 4 \times 2.25 = 9 \).
Result: \( I = \dfrac{9}{d^2} \). A1
[2 marks]
Sound intensity varies inversely with the square of distance: \( I = \dfrac{k}{d^2} \). M1
Given: \( I = 4 \) when \( d = 1.5 \). Substitute: \( 4 = \dfrac{k}{1.5^2} \).
Calculate: \( 1.5^2 = 2.25 \), so \( 4 = \dfrac{k}{2.25} \), \( k = 4 \times 2.25 = 9 \).
Result: \( I = \dfrac{9}{d^2} \). A1
[2 marks]
(b)
Function: \( I = \dfrac{9}{d^2} \), inverse square relationship. A1
Plot point: (1.5, 4). Shape: Hyperbolic decay, asymptotic to \( d \)-axis (\( I \to 0 \) as \( d \to \infty \)) and \( I \)-axis (\( I \to \infty \) as \( d \to 0^+ \)).
Curve: Smooth, decreasing, passing through (1.5, 4). A1
[2 marks]
Function: \( I = \dfrac{9}{d^2} \), inverse square relationship. A1
Plot point: (1.5, 4). Shape: Hyperbolic decay, asymptotic to \( d \)-axis (\( I \to 0 \) as \( d \to \infty \)) and \( I \)-axis (\( I \to \infty \) as \( d \to 0^+ \)).
Curve: Smooth, decreasing, passing through (1.5, 4). A1
[2 marks]
(c)
Siren audible if \( I > 1.5 \times 10^{-6} \). Inaudible if \( \dfrac{9}{d^2} \leq 1.5 \times 10^{-6} \). M1
Solve: \( \dfrac{9}{d^2} \leq 1.5 \times 10^{-6} \), so \( d^2 \geq \dfrac{9}{1.5 \times 10^{-6}} = 6 \times 10^6 \).
Square root: \( d \geq \sqrt{6 \times 10^6} \approx 2449.48 \).
Rounded: \( d \geq 2450 \, \text{m} \). A1
[2 marks]
Siren audible if \( I > 1.5 \times 10^{-6} \). Inaudible if \( \dfrac{9}{d^2} \leq 1.5 \times 10^{-6} \). M1
Solve: \( \dfrac{9}{d^2} \leq 1.5 \times 10^{-6} \), so \( d^2 \geq \dfrac{9}{1.5 \times 10^{-6}} = 6 \times 10^6 \).
Square root: \( d \geq \sqrt{6 \times 10^6} \approx 2449.48 \).
Rounded: \( d \geq 2450 \, \text{m} \). A1
[2 marks]
Total Marks: 6