Question

1. the alarm in a noisy factory is a siren whose volume, v decibels, fluctuates so that t seconds after starting, the volume is given by the sinusoidal function: ( v(t)=18 sin left(\frac{pi}{15} t

ight)+60 ).
a) what are the maximum and minimum volumes of the siren? (1 mark)
b) determine the period of the function. (1 mark)

Explanation:

Step1: Find maximum and minimum values

For the function \(y = A\sin(Bt)+C\), the maximum value is \(C + |A|\) and the minimum value is \(C-|A|\). Here \(A = 18\), \(C=60\).
Maximum value: \(60 + 18=78\)
Minimum value: \(60-18 = 42\)

Step2: Calculate the period

The formula for the period of \(y=\sin(Bt)\) is \(T=\frac{2\pi}{B}\). For \(V(t)=18\sin(\frac{\pi}{15}t)+60\), \(B=\frac{\pi}{15}\)
\(T=\frac{2\pi}{\frac{\pi}{15}}=2\pi\times\frac{15}{\pi}=30\)

Answer:

a) The maximum volume is \(78\) decibels and the minimum volume is \(42\) decibels.
b) The period of the function is \(30\) seconds.