physics
posted by sarah .
Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 172 \rm Hz. You are 8.00 \rm m from speaker A. Take the speed of sound in air to be 344 \rm m/s.
What is the closest you can be to speaker B and be at a point of destructive interference?

draw the figure. draw a triangle short side from A, long side from B, other side A to B.
so the difference between the short side and the long side has to be an even multiple of 1/2 wavelength.
start with the wave equation
lambda= v/f
so long sideshort side= n(v/2f)
do the degenerate triangle first, where short+long= AB (a point on AB)
nv/2f+nv/2f+v/2f=ab
or v/2f(2n+1)=ab
you know ab as 8m, so solve for n
then solve for short side length.
Now do the same thinking for some triangle .
Use the law of cosines to find short side, then minimize it with differential calculus.
You will find the answer, and discovering something magic about the angle from B, it gets minimized also.