When cleaning up a local beach, students found many different particles that were in the water that affected the shoreline, like
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Answer:
Approximately .
Explanation:
.
.
Notice that sine waves and
share the same frequency and wavelength. The only distinction between these two waves is the
in
.
Therefore, the sum would still be a sine wave. The amplitude of
could be found without using calculus.
Consider the sum-of-angle identity for sine:
.
Compare the expression to
. Let
and
. Apply the sum-of-angle identity of sine to rewrite
.
.
Therefore, the sum would become:
.
Consider: would it be possible to find and
that satisfy the following hypothetical equation?
.
Simplify this hypothetical equation:
.
Apply the sum-of-angle identity of sine to rewrite the left-hand side:
.
Compare this expression with the right-hand side. For this hypothetical equation to hold for all real and
, the following should be satisfied:
, and
.
Consider the Pythagorean identity. For any real number :
.
Make use of the Pythagorean identity to solve this system of equations for . Square both sides of both equations:
.
.
Take the sum of these two equations.
Left-hand side:
.
Right-hand side:
.
Therefore:
.
.
Substitute back to the system to find
. However, notice that the exact value of
isn't required for finding the amplitude of
.
(Side note: one possible value of is
radians.)
As long as is a real number, the amplitude of
would be equal to the absolute value of
.
Therefore, the amplitude of would be:
.