IB Math AI HL  Popular Quizzes
Complex Numbers & Sine Waves
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Question 1
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hard
[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.

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]

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

Show that the profits of the hotel can be modelled by the function $\mathrm{P}(t) = 83.9\sin\hspace{0.05em}(0.0172t1.51) + 119$. [3]

According to the model, find:

the highest profit the hotel will make;

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

hard
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Question 3
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hard
[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.

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]

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

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