1. (25 Points) Laura, the coordinator of a music festival,...

1. (25 Points) Laura, the coordinator of a music festival, predicts that if she advertises the festival x days before it occurs, the expected income will be R(x) thousand dollars, where: $R(x) = 400 + 120x - x^2$ The expenses for promoting the festival for x days amount to C(x) thousand dollars, where: $C(x) = 2x^2 + 300$ a. Develop the profit function. b. Determine the ideal number of days Laura should promote the festival in advance to maximize her profit. Also, calculate the maximum profit and show that this point is indeed a maximum. c. Write the expression for the ratio of income to expenses and analyze the behavior of this ratio as x approaches 0. d. Explore how the ratio of income to expenses changes as x approaches infinity.

Transcript

00:01 Alright, let's take a look at a problem where somebody is organizing a big event.

00:06 And for every day that they announce the event in advance of when the event actually happens, they earn additional revenue, which is modeled by this equation, r of x equals 400 plus 120x minus x squared.

00:26 However, the cost of advertising that event prior to an event happens is modeled by this c of x function, 2x squared plus 300.

00:36 So just to talk about what each of these mean, the first thing is in this revenue function, this 400 is like guaranteed money.

00:45 This is the money they would get with no advance warning.

00:56 And 120x minus x squared is this inverse parabola of revenue.

01:03 So it actually will max out at a certain time.

01:07 You get the most possible revenue and then eventually decrease.

01:10 So it wouldn't be like if you told people about this event 10 years in advance that they would actually get more revenue.

01:17 That certainly would not make any sense.

01:18 For the cost function, it's similar.

01:21 This 300 is the guaranteed costs.

01:27 So this is how much is going to cost them, again, with no advance warning.

01:32 So if they just said, okay, everybody, like we're having the event, here we go, it costs $300.

01:37 They would make $400, but it costs $300.

01:41 Different from the revenue function, for each day that they announce this event beforehand, it actually increases the cost by two times a number of days squared.

01:53 And that will go on exponentially.

01:57 So if you announce the event 10 years in advance, you're going to be spending way more money than what you would want to spend.

02:05 All right.

02:06 So let's look at a couple different things here.

02:08 The first is the profit function.

02:10 The profit function is defined by p of x, and it is just this real.

02:18 Revenue function minus the cost function, just the difference between the two.

02:23 So our revenue is 400 plus 120x minus x squared, and we're subtracting our cost function.

02:34 So the cost function again is 2x squared plus 300.

02:40 And now we can just combine some like terms.

02:42 So we'll distribute this negative.

02:45 It'll be 400 and it'll be a negative 300.

02:49 So i'll just bring that over here, minus 300, plus 120x, minus x squared, and then minus 2x squared.

03:00 So again, from the cost function, this minus 300 is the cost and minus 2x squared.

03:07 And you can already start to see like that profit piece that we were talking about earlier, 400 minus 300.

03:13 Here's your guaranteed $100 if you did it with no advance warning.

03:18 So there's 100 plus 120x minus 3x squared.

03:27 Now when thinking about this, if you want to graph this, this is not the best form to use.

03:35 It's helpful that know the y intercept, which is that 100.

03:38 So you can say, okay, i know my y intercept is 0 ,100.

03:41 But you probably want to put this into a vertex form.

03:45 And in vertex form, we're just going to move some things around.

03:49 So our profit is equal to.

03:51 I'm just going to put our in a descending order of exponents.

03:58 So negative 3x squared plus 120x plus 100.

04:05 This would be really helpful in vertex form.

04:07 Then we would know what the maximum value is, where this inverted parabola.

04:12 Turns around, et cetera.

04:14 So in order to do this, i'm going to focus on these first two terms, the negative 3x squared plus 120x, because in vertex form, i'm going to have to complete the square.

04:29 So let me pull out this negative three, factored out of those first two terms, and i get x squared and then minus 40x, because negative 40x times negative 3 is positive.

04:42 120x.

04:44 And then the 100 is still just there.

04:46 Again, this lines up with what we talked about earlier in the profit.

04:50 If you put in zero days, no advance warning, you're still getting $100 profit.

04:54 All right.

04:55 To complete the square, we have to look at this little grouping in here and take the second term, this b term, the negative 40, and we're going to divide it by two and then square it.

05:12 So adding that in, we'll get negative three times x squared minus 40x divided by 2 is negative 20, and negative 20 squared is positive 400 plus 100.

05:30 Now, you can't just add something in for fun.

05:33 If you do it on the right hand side of the equation, then you better do it on the left.

05:36 So remember, this is still, you know, like our profit function.

05:39 But we got to add in that's 400...

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