The average monthly precipitation p in inches including rain...

The average monthly precipitation P in inches, including rain, snow, and ice, for a city can be modeled by P = 1.09sin(0.59t + 3.94) + 1.63, 0 ≤ t ≤ 12, where t is the time in months, with t = 1 corresponding to January. (Round your answers to 2 decimal places.) (a) Find the maximum and minimum precipitation and when it occurs. A maximum of inches occurs at t = ___ A minimum of inches occurs at t = 1.309 and t = ___ (Enter your answers from smallest to largest.) (b) Determine the average monthly precipitation for the year. =___ inches/month (c) Find the total annual precipitation for the city =___ inches

Transcript

00:01 Okay, in this question we have been given this precipitation function, which is the average monthly precipitation in inches for a city.

00:12 Now what we want to do is begin by finding the maximum and minimum precipitation and when it occurs.

00:18 And what's nice is that we don't have to use calculus techniques to do this if you recognise that sine is biggest at 1 and smallest at minus 1.

00:28 So the maximum is going to occur when this sine function, when say sine of x equals 1.

00:37 And sine of x equals 1 when x is pi over 2, 5 pi over 2, and so on.

00:44 You can just keep adding 2 pi.

00:48 So sine of x equals 1 at these points and x in this case is 0 .59t plus 3 .94.

00:57 So what happens when 0 .59t plus 3 .94 equals pi over 2? what value of t do we get? and you should get a value of t of minus 4 .016.

01:12 And we're going to disregard this because that's negative time, we aren't interested in that.

01:17 So let's look at the next solution.

01:18 What happens when 0 .59t plus 3 .94 equals 5 pi over 2? what value of t does that correspond to? and you should find that you get a value of t of 6 .633867 and you can round that however you like.

01:38 Okay so the maximum occurs when t takes on this value.

01:42 And what's the precipitation when t takes on that value? let's call it p max.

01:48 Okay well we don't actually have to put in this value of t into the original equation because we know already that sine is 1 there.

01:56 So p max is just 1 .09 times 1 plus 1 .63.

02:02 So if you type that in, 1 .09 plus 1 .63, you should get 2 .72.

02:15 Okay so that is your maximum and now let's consider your minimum.

02:20 Now we're going to use the same idea here, remembering that sine of x has a minimum at minus 1.

02:27 And that happens when x is 3 pi over 2 and also when x is 7 pi over 2...