A sound wave can be modelled using a function \(P(t)=k\, \sin a t\), where \(P\) is air pressure in Pascals, \(t\) is time in milliseconds (ms) and \(k\) and \(a\) are constants.
- Write the equation for a sound wave \(P_1(t)\) that has an amplitude of 2 Pascals and a period of 5 ms. (2 marks)
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- The graph of \(P_1(t)\) from part (a) is shown.
- On the diagram, sketch the graph of \(P_2(t)=4 \sin \left(\dfrac{\pi}{10} t\right)\) for \(0 \leq t \leq 10\). (2 marks)
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- Hence, find the values of \(t\), where \(0<t<10\), for which functions \(P_1(t)\) and \(P_2(t)\) are BOTH decreasing. (2 marks)
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a. \(P_1(t)=2\, \sin \left(\dfrac{2 \pi t}{5}\right)\)
b.
c. \(\text{Both decreasing for} \ \ 6.25<t<8.75\)
a. \(\text{Amplitude}=2 \Rightarrow k=2\)
\(\text{Period}=5\)
| \(\dfrac{2 \pi}{a}\) | \(=5\) |
| \(5a\) | \(=2 \pi\) |
| \(a\) | \(=\dfrac{2 \pi}{5}\) |
\(\therefore P_1(t)=2\, \sin \left(\dfrac{2 \pi t}{5}\right)\)
b.
\(P_2(t)=4\, \sin \left(\dfrac{\pi}{10} t\right)\)
\(\text{Amplitude}=4\)
\(\text{Period}=\dfrac{2 \pi}{\frac{\pi}{10}}=20\)
c. \(\text {By inspection of graph:}\)
\(P_2(t) \ \text {is decreasing for} \ \ 5<t \leq10\)
\(P_1(t) \text { is decreasing for} \ \ 1.25<t<3.75 \ \ \text{and}\ \ 6.25<t<8.75\)
\(\therefore \ \text{Both decreasing for} \ \ 6.25<t<8.75\)