In a certain city, the number of hours of sunlight on the summer solstice is 15.85 O and the number of hours of sunlight on the winter solstice is 8.340. Answer the following questions. Assume that summer solstice occurs on the 172nd day of the year and that there are 365 days until the next summer solstice. So what the 1585 O is really representing is it's the maximum value of our range, and what the 8.340 is really representing is the minimum of the range. So when it wants us to find the amplitude that a the way we find the amplitude is we take the maximum and we subtract the minimum and then we divide it by two. So maximum minus minimum divided by two is going to give us 3.755. The next thing it's asking for us to do is to find the Omega. Well, our Omega is going to be 2π over Omega is the period, and in this case the periods 365 S To solve this, we'd have 2π / 365 equaling our Omega. The next piece is asking us for the Phi. These are Greek letters. You could have used ABCD, whatever you want, but this one's Phi. So the 323π / 730, this one's a little harder to come across. And what we're going to have to do is we're going to have to think about where the regular height of a sine curve occurs versus where it's going to occur for our summer solstice. So if this was 365 days, we know that it's split up into four increments. So this here would really be 91.25 if I took the 365 and divided it by 4. Now we're being told though, that our height is actually going to occur at the 172nd day. So our height is going to get moved. It's going to transform to over here somewhere. So if we take 172 and we subtract that 91.25, that should tell us what this distance is in between here and this is 80.75. So now what we need to do is we need to take the 2π / 365 and multiply it by 80 3/4. That would be 4 * 80 is 320 thirds over 4. So if we have 323 / 4, the four and the two will reduce to A2 down here. So we get 323π / 730. The B this B here is just our vertical translation. And the way we get our vertical translation is instead of subtracting the Max and min, we're going to add the Max and min and divide by two. So 15.850 + 8.340 / 2, we're finding the average here. So that's going to be that one. Now, if we put it all together in an equation, our equation's going to be Y equal 3.755 sine 2π / 365 * X -, 323π / 730 plus 12.095. So when it asks what happens at the 91st day, literally all we're doing is we're putting in 91 for the X and solving, and you're going to get 12.75 hours. So to finish this problem up, we have a graph. And if we look at our graph, we know that on our 91st day, we're going to be 12.75, and we know on our 172nd day we're at the highest we can be. So looking at this graph and realizing that they're counting by 50s, fifty a hundred 150, a little bit less than 100, we should have 12.75 for our 91. That would look about right. And then at our 171st day, a little bit before 175, we should be at our absolute Max. And we are in this graph.