00:01
Okay, so in this question we have a microphone placed right in the centre of the picture there, and the microphone is placed four metres in front of the stage, so we know that this distance that i've just marked is four metres.
00:18
Okay, and we've been given the equation for the optimal sound pickup region, which is r equals eight plus eight sine theta.
00:31
And what we want to know is what is the area that's shaded in yellow.
00:38
So we are going to need to compute the area of the polar graph between two half lines, theta equals alpha, theta equals beta.
00:52
So eventually we're going to be computing this, the area between alpha and beta of a half r squared d theta, because that's how we find area with polar coordinates.
01:03
So the two half lines, theta equals alpha and theta equals beta, are going to be given by ones that i'll draw right now for you.
01:10
So there's one half line, and there is the other half line.
01:17
And alpha is measured from the positive x -axis, so that's alpha there.
01:22
And likewise, beta is also measured from the positive x -axis, so it's going to be a lot bigger.
01:29
Okay, and then once we've done that, we're going to subtract off the area of the isosceles triangle that we are not interested in for the total area.
01:42
So we're going to need to find out the length of the solid green line, as well as alpha itself.
01:50
So in order to do that, we need to work out what this coordinate is here.
01:54
So what do we know about this coordinate? well, we definitely don't know its x -coordinate, but we do know its y -coordinate is going to be 4, measured from the origin, which is right there.
02:07
And remember, 4 can also be written, or y can also be written, as r sin theta.
02:15
And we know what r is, that's 8 plus 8 sin theta.
02:20
So that means that 4 equals 8 plus 8 sin theta, all multiplied by sin theta.
02:31
And so we get 4 equals 8 sin theta plus 8 sin squared theta, okay? and if we divide through by 4 and we subtract off the 1 to the other side, we get 2 sin squared theta plus 2 sin theta minus 1 equals 0.
02:58
And this, in fact, is just a quadratic, and you can solve this using the quadratic formula.
03:05
But what you'll find is that sin theta equals minus 1 plus or minus the square root of root 3, and that's all divided by 2.
03:18
Okay, so we can do sin minus 1 of each of these to find out what our angles alpha and beta are going to be.
03:26
So minus 1 plus root 3, all divided by 2, take sin minus 1, and we want to be working in radians, remember.
03:33
We get theta equals 0 .3747.
03:39
So this is my alpha.
03:41
And then if we do the same with the negative root, what we find is a math error.
03:47
So the negative root isn't going to help us here, and instead, what we need to consider is the graph of sin to find the other value of theta.
03:55
So considering the graph, what you have to do is pi minus 0 .3747, and if you do that, you get 2 .7669, and that's where i've rounded...