Question

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

• A. 3 m
• B. 2 m
• C. 1 m
• D. 5 m
• E. 4 m

Hint:

When combining sinusoidal waves, the total amplitude can be found using the Pythagorean theorem \( A_{\text{total}} = \sqrt{A_1^2 + A_2^2} \), where \( A_1 \) and \( A_2 \) are the individual amplitudes.

Answer 1

Answer: (E)

The given wave equation is: \[ y = 3\sin(3x - 5t) + A\cos(3x - 5t) \] This is a combination of a sine and cosine wave, where the total amplitude \( A_{\text{total}} \) can be found using the following formula for the resultant amplitude when combining sinusoidal functions of the same frequency: \[ A_{\text{total}} = \sqrt{(3)^2 + (A)^2} \] We are given that the total amplitude is 5 m: \[ 5 = \sqrt{9 + A^2} \] Squaring both sides: \[ 25 = 9 + A^2 \] \[ A^2 = 16 \] \[ A = 4 \text{ m} \] Thus, the value of \( A \) is 4 m.