[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:06.13,Default,,0000,0000,0000,,Suppose marketing experts have determined the relationship between the selling price of an item
Dialogue: 0,0:00:06.13,0:00:14.45,Default,,0000,0000,0000,,and the cost of an item can be represented by the linear equation q = -30s + 800,
Dialogue: 0,0:00:14.45,0:00:19.09,Default,,0000,0000,0000,,where q is the quantity sold in a year and s is the selling price.
Dialogue: 0,0:00:19.09,0:00:23.75,Default,,0000,0000,0000,,If the cost to produce the item is $20,
Dialogue: 0,0:00:23.75,0:00:27.51,Default,,0000,0000,0000,,so the cost to produce an item is $20,
Dialogue: 0,0:00:27.51,0:00:33.15,Default,,0000,0000,0000,,what is the selling price that optimizes the yearly, the yearly, profit?
Dialogue: 0,0:00:33.15,0:00:36.33,Default,,0000,0000,0000,,So what's the profit going to be? So let me write this down.
Dialogue: 0,0:00:36.33,0:00:42.85,Default,,0000,0000,0000,,So a yearly profit is going to be the quantity, is going to be, the quantity that we sell in a year,
Dialogue: 0,0:00:42.85,0:00:45.82,Default,,0000,0000,0000,,it's going to be the quantity that we sell in a year,
Dialogue: 0,0:00:45.82,0:00:53.35,Default,,0000,0000,0000,,times, times, the price that we sell it at, price, times the price that we sell it at,
Dialogue: 0,0:00:53.35,0:01:00.25,Default,,0000,0000,0000,,minus the cost of us actually producing that item, and in this case they tell us it is $20.
Dialogue: 0,0:01:00.25,0:01:05.87,Default,,0000,0000,0000,,So for example, if we sell two items, if q is 2 and we sell them for $25,
Dialogue: 0,0:01:05.87,0:01:11.07,Default,,0000,0000,0000,,we're gonna make $5 on each item, 'cause it cost each of them cost us 5, $20 to produce,
Dialogue: 0,0:01:11.07,0:01:13.74,Default,,0000,0000,0000,,so 25 minus 20 will be 5.
Dialogue: 0,0:01:13.74,0:01:20.10,Default,,0000,0000,0000,,If we sell two items at that price it'll be 2 times 5 or we'll have a profit of $10.
Dialogue: 0,0:01:20.10,0:01:23.48,Default,,0000,0000,0000,,So what, how can we figure out how to maximize this profit?
Dialogue: 0,0:01:23.48,0:01:27.56,Default,,0000,0000,0000,,Well they gave us the quantity as a function of selling price,
Dialogue: 0,0:01:27.56,0:01:31.68,Default,,0000,0000,0000,,so we could, we could express the entire profit as a function of selling price.
Dialogue: 0,0:01:31.68,0:01:42.74,Default,,0000,0000,0000,,So we could say, we can substitute q is equal to -30 s plus 800 right over, right over here.
Dialogue: 0,0:01:42.74,0:01:44.64,Default,,0000,0000,0000,,And let's be very clear what this is telling us.
Dialogue: 0,0:01:44.64,0:01:50.72,Default,,0000,0000,0000,,This is telling us that if the selling price increases, then this will become a larger negative number
Dialogue: 0,0:01:50.72,0:01:55.07,Default,,0000,0000,0000,,so we're going to sell fewer, we're gonna sell a smaller quantity.
Dialogue: 0,0:01:55.07,0:01:58.40,Default,,0000,0000,0000,,And actually if you believe this, and if you actually made the selling price zero,
Dialogue: 0,0:01:58.40,0:02:04.12,Default,,0000,0000,0000,,if you just gave away this product, it tells us that we would sell at most 800.
Dialogue: 0,0:02:04.15,0:02:07.06,Default,,0000,0000,0000,,So it might not be a perfect model but let's just use this for,
Dialogue: 0,0:02:07.06,0:02:10.55,Default,,0000,0000,0000,,you know some marketing experts have told us this, so let's just use it.
Dialogue: 0,0:02:10.55,0:02:21.94,Default,,0000,0000,0000,,So if we substitute -30 s plus 800 for q, we get -30 s plus 800 times s minus 20,
Dialogue: 0,0:02:21.94,0:02:26.80,Default,,0000,0000,0000,,times, and this is in a different shade of yellow, times s minus 20.
Dialogue: 0,0:02:26.80,0:02:30.73,Default,,0000,0000,0000,,This is profit as a function of selling price.
Dialogue: 0,0:02:30.73,0:02:34.65,Default,,0000,0000,0000,,And now we can just, let me be very careful here, let me be very care--
Dialogue: 0,0:02:34.65,0:02:38.31,Default,,0000,0000,0000,,This is just, this is q right here, and so this whole thing is q.
Dialogue: 0,0:02:38.31,0:02:43.01,Default,,0000,0000,0000,,Wanna make sure we're multiplying this whole expression times this entire expression right over there,
Dialogue: 0,0:02:43.01,0:02:44.69,Default,,0000,0000,0000,,and so let's do that.
Dialogue: 0,0:02:44.69,0:02:47.39,Default,,0000,0000,0000,,So this is going to be equal to,
Dialogue: 0,0:02:47.44,0:02:50.28,Default,,0000,0000,0000,,this is going to be equal to -30 s.
Dialogue: 0,0:02:50.28,0:02:54.31,Default,,0000,0000,0000,,So let me just distribute it out. This is going to be -30 s,
Dialogue: 0,0:02:54.31,0:02:59.20,Default,,0000,0000,0000,,times s minus 20, times this whole thing, we're taking this whole term,
Dialogue: 0,0:02:59.20,0:03:01.59,Default,,0000,0000,0000,,we're first multiplying it times -30 s.
Dialogue: 0,0:03:01.59,0:03:07.39,Default,,0000,0000,0000,,And then we're gonna take this whole term and then multiply it by 800, s minus 20.
Dialogue: 0,0:03:07.39,0:03:11.99,Default,,0000,0000,0000,,And so this gives us, this is equal to -30 s times s, we have to distribute again,
Dialogue: 0,0:03:11.99,0:03:24.90,Default,,0000,0000,0000,,-30 s squared, -30 s times -20, is going to be positive, positive, positive 600 s.
Dialogue: 0,0:03:24.90,0:03:29.18,Default,,0000,0000,0000,,And then we have 800 times s, so that's plus 800 s,
Dialogue: 0,0:03:29.18,0:03:36.49,Default,,0000,0000,0000,,and then 800 times -20, so that is -8 times 2 is 16,
Dialogue: 0,0:03:36.49,0:03:40.60,Default,,0000,0000,0000,,and we have one, two, three zeros, one, two, three, zeros.
Dialogue: 0,0:03:40.60,0:03:44.22,Default,,0000,0000,0000,,And if we simplify we can add these two terms right over here.
Dialogue: 0,0:03:44.22,0:03:52.85,Default,,0000,0000,0000,,We get -30 s squared plus 1400 s minus 16000.
Dialogue: 0,0:03:52.85,0:03:59.51,Default,,0000,0000,0000,,So we now, we've now expressed, we've now expressed, our profit as a function of selling price.
Dialogue: 0,0:03:59.51,0:04:01.98,Default,,0000,0000,0000,,And this is actually going to be a downward opening parabola,
Dialogue: 0,0:04:01.98,0:04:06.33,Default,,0000,0000,0000,,and we can tell that because the coefficient on the second degree term,
Dialogue: 0,0:04:06.33,0:04:08.36,Default,,0000,0000,0000,,on the quadratic term, is negative.
Dialogue: 0,0:04:08.36,0:04:13.69,Default,,0000,0000,0000,,So if we were to graph this, if we were to graph this...
Dialogue: 0,0:04:13.69,0:04:17.65,Default,,0000,0000,0000,,So over here--let me draw a better graph than that.
Dialogue: 0,0:04:17.65,0:04:20.51,Default,,0000,0000,0000,,Over here, this axis right here is going to be the selling price,
Dialogue: 0,0:04:20.51,0:04:22.97,Default,,0000,0000,0000,,And this is profit which is a function of selling price.
Dialogue: 0,0:04:22.97,0:04:29.77,Default,,0000,0000,0000,,This graph, this equation right over here, is going to look like this, is going to look something like this.
Dialogue: 0,0:04:29.77,0:04:33.75,Default,,0000,0000,0000,,We already saw the selling price--let me write, just write it this way.
Dialogue: 0,0:04:33.75,0:04:38.98,Default,,0000,0000,0000,,So let me just--is going to look something like this.
Dialogue: 0,0:04:38.98,0:04:42.64,Default,,0000,0000,0000,,I don't know what the exact equation is gonna look like, but it's gonna be downward opening.
Dialogue: 0,0:04:42.64,0:04:46.16,Default,,0000,0000,0000,,And what we wanna do is maximize the profit.
Dialogue: 0,0:04:46.16,0:04:49.42,Default,,0000,0000,0000,,We wanna find this maximum point right over here.
Dialogue: 0,0:04:49.42,0:04:54.29,Default,,0000,0000,0000,,You could do it with calculus, if you had, if you had calculus at your, at your, at your fingertips.
Dialogue: 0,0:04:54.29,0:04:57.75,Default,,0000,0000,0000,,Or you could just recognize this is the vertex of the parabola.
Dialogue: 0,0:04:57.75,0:05:01.18,Default,,0000,0000,0000,,And you could, you could figure out the vertex by putting in the vertex form,
Dialogue: 0,0:05:01.18,0:05:04.98,Default,,0000,0000,0000,,but the fastest way is to just know that the normally the x coordinate,
Dialogue: 0,0:05:04.98,0:05:11.23,Default,,0000,0000,0000,,or the s coordinate, the s coordinate, of the vertex is going to be -b over 2 a.
Dialogue: 0,0:05:11.23,0:05:13.94,Default,,0000,0000,0000,,And if we wanna figure out what -b over 2 a is,
Dialogue: 0,0:05:13.94,0:05:18.76,Default,,0000,0000,0000,,we just take the--this is the b right over here so it's a -b
Dialogue: 0,0:05:18.76,0:05:26.39,Default,,0000,0000,0000,,so it's -1400 over 2 a, over 2 times -30,
Dialogue: 0,0:05:26.39,0:05:32.58,Default,,0000,0000,0000,,which is equal to -1400 over -60.
Dialogue: 0,0:05:32.58,0:05:36.51,Default,,0000,0000,0000,,Negatives cancel out, we could divide the numerator and the denominator by 10.
Dialogue: 0,0:05:36.51,0:05:40.21,Default,,0000,0000,0000,,So this is the same thing as a 140 over 6.
Dialogue: 0,0:05:40.21,0:05:46.35,Default,,0000,0000,0000,,We can divide the numerator and the denominator by 3, or by, by 2.
Dialogue: 0,0:05:46.35,0:05:51.57,Default,,0000,0000,0000,,And you get 70, you get 70 over 3.
Dialogue: 0,0:05:51.74,0:05:56.91,Default,,0000,0000,0000,,And then we can just divide that, so 3 goes into 70,
Dialogue: 0,0:05:56.91,0:06:00.54,Default,,0000,0000,0000,,3 goes into 7 two times, 2 times 3 is 6.
Dialogue: 0,0:06:00.54,0:06:06.30,Default,,0000,0000,0000,,Subtract, you get a difference of 1, bring down the zero, 3 goes into 10 three times,
Dialogue: 0,0:06:06.30,0:06:11.01,Default,,0000,0000,0000,,3 times 3 is 9, subtract, you have, bring down, you get a 1.
Dialogue: 0,0:06:11.01,0:06:15.07,Default,,0000,0000,0000,,Now we're in the decimals, we bring down another zero, it becomes a 10 again,
Dialogue: 0,0:06:15.13,0:06:18.62,Default,,0000,0000,0000,,3 goes into 10 three times, I think you see where this is going.
Dialogue: 0,0:06:18.62,0:06:25.66,Default,,0000,0000,0000,,It's 23.3 repeating times. If we just keep doing this, we'll just keep getting more, more, more threes.
Dialogue: 0,0:06:25.66,0:06:30.14,Default,,0000,0000,0000,,Or if we just wanted to round to the nearest penny, since we're talking about selling something,
Dialogue: 0,0:06:30.14,0:06:38.75,Default,,0000,0000,0000,,this optimal profit, this optimal profit, will happen at a selling price of $23 and 30,
Dialogue: 0,0:06:38.75,0:06:42.33,Default,,0000,0000,0000,,$23 and 33 cents.
Dialogue: 0,0:06:42.33,9:59:59.99,Default,,0000,0000,0000,,That will optimize the yearly profit.