Discussion

If the amplitude of the wave \(y = 3sin(3x - 5t) + Acos(3x - 5t)\) m is 5 m, the value of A is

Answer: E

\(A = \) 4 m.

A sine and a cosine of the same phase are 90° apart, so they combine like perpendicular vectors:

\[ \begin{aligned} R &= \sqrt{3^{2} + A^{2}} = 5 \\ A^{2} &= 25 - 9 = 16,\quad A = 4\ \text{m} \end{aligned} \]

2 m comes from simply subtracting amplitudes (5 - 3), which would be right only for waves in phase.

Formally, \(3\sin\theta + A\cos\theta = R\sin(\theta + \delta)\) with \(R = \sqrt{9 + A^{2}}\), which is the resultant amplitude.

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