Complex Numbers & Sine Waves

[Maximum mark: 5]

Two voltage sources are connected to a circuit. At time $t$ milliseconds (ms), the voltage from the first source is $V_1(t) = 12\cos(20t)$ and the voltage from the second source is $V_2(t) = 18\cos(20t+5)$, where both $V_1(t)$ and $V_2(t)$ are measured in volts.

1. Write, in the form $V(t) = A\cos\hspace{0.05em}(\omega t+\varphi)$, an expression for the total voltage in the circuit at time $t$ ms. [4]
   

2. Hence write down the highest voltage in the circuit. [1]
   

[Maximum mark: 6]

The revenues of a four seasons hotel can be modelled by the function

$\mathrm{R}(t) = 58.2\sin\hspace{0.05em}(0.0172t - 1.25) + 204$,

where $t$ is the number of days after midnight on $31$ December.

In a similar way, the operating costs of the hotel can be modelled by the function

$\mathrm{C}(t) = 31.4\sin\hspace{0.05em}(0.0172t + 1.14) + 85.0$.

Both $\mathrm{R}(t)$ and $\mathrm{C}(t)$ are measured in thousand dollars.

1. Show that the profits of the hotel can be modelled by the function $\mathrm{P}(t) = 83.9\sin\hspace{0.05em}(0.0172t-1.51) + 119$. [3]
   

2. According to the model, find:

   1. the highest profit the hotel will make;

   2. the date on which the highest profit will occur. [3]
      

[Maximum mark: 6]

Ali is swimming in a public pool with some of his friends. At time $t$ seconds, he $\text{encounters}$ some waves with height $h_1(t) = 0.15\sin\hspace{0.05em}(3t)$ from big Bobby jumping into the pool, and waves of height $h_2(t) = 0.08\sin\hspace{0.05em}(3t+1.25)$ from small Suzie jumping into the pool. Both $h_1(t)$ and $h_2(t)$ are measured in metres.

1. Write, in the form $h(t) = A\sin\hspace{0.05em}(\omega t+\varphi)$, an expression for the total height of the waves Ali encounters at time $t$ seconds. [3]
   

2. Find the times in the first $5$ seconds when Ali isn't affected by any waves. [2]
   

3. Find the first time when the waves reaching Ali has maximum height. [1]