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Suppose marketing experts have determined the relationship between the selling price of an item

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and the cost of an item can be represented by the linear equation q = -30s + 800,

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where q is the quantity sold in a year and s is the selling price.

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If the cost to produce the item is $20,

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so the cost to produce an item is $20,

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what is the selling price that optimizes the yearly, the yearly, profit?

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So what's the profit going to be? So let me write this down.

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So a yearly profit is going to be the quantity, is going to be, the quantity that we sell in a year,

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it's going to be the quantity that we sell in a year,

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times, times, the price that we sell it at, price, times the price that we sell it at,

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minus the cost of us actually producing that item, and in this case they tell us it is $20.

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So for example, if we sell two items, if q is 2 and we sell them for $25,

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we're gonna make $5 on each item, 'cause it cost each of them cost us 5, $20 to produce,

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so 25 minus 20 will be 5.

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If we sell two items at that price it'll be 2 times 5 or we'll have a profit of $10.

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So what, how can we figure out how to maximize this profit?

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Well they gave us the quantity as a function of selling price,

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so we could, we could express the entire profit as a function of selling price.

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So we could say, we can substitute q is equal to -30 s plus 800 right over, right over here.

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And let's be very clear what this is telling us.

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This is telling us that if the selling price increases, then this will become a larger negative number

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so we're going to sell fewer, we're gonna sell a smaller quantity.

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And actually if you believe this, and if you actually made the selling price zero,

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if you just gave away this product, it tells us that we would sell at most 800.

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So it might not be a perfect model but let's just use this for,

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you know some marketing experts have told us this, so let's just use it.

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So if we substitute -30 s plus 800 for q, we get -30 s plus 800 times s minus 20,

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times, and this is in a different shade of yellow, times s minus 20.

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This is profit as a function of selling price.

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And now we can just, let me be very careful here, let me be very care--

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This is just, this is q right here, and so this whole thing is q.

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Wanna make sure we're multiplying this whole expression times this entire expression right over there,

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and so let's do that.

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So this is going to be equal to,

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this is going to be equal to -30 s.

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So let me just distribute it out. This is going to be -30 s,

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times s minus 20, times this whole thing, we're taking this whole term,

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we're first multiplying it times -30 s.

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And then we're gonna take this whole term and then multiply it by 800, s minus 20.

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And so this gives us, this is equal to -30 s times s, we have to distribute again,

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-30 s squared, -30 s times -20, is going to be positive, positive, positive 600 s.

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And then we have 800 times s, so that's plus 800 s,

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and then 800 times -20, so that is -8 times 2 is 16,

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and we have one, two, three zeros, one, two, three, zeros.

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And if we simplify we can add these two terms right over here.

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We get -30 s squared plus 1400 s minus 16000.

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So we now, we've now expressed, we've now expressed, our profit as a function of selling price.

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And this is actually going to be a downward opening parabola,

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and we can tell that because the coefficient on the second degree term,

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on the quadratic term, is negative.

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So if we were to graph this, if we were to graph this...

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So over here--let me draw a better graph than that.

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Over here, this axis right here is going to be the selling price,

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And this is profit which is a function of selling price.

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This graph, this equation right over here, is going to look like this, is going to look something like this.

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We already saw the selling price--let me write, just write it this way.

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So let me just--is going to look something like this.

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I don't know what the exact equation is gonna look like, but it's gonna be downward opening.

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And what we wanna do is maximize the profit.

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We wanna find this maximum point right over here.

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You could do it with calculus, if you had, if you had calculus at your, at your, at your fingertips.

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Or you could just recognize this is the vertex of the parabola.

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And you could, you could figure out the vertex by putting in the vertex form,

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but the fastest way is to just know that the normally the x coordinate,

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or the s coordinate, the s coordinate, of the vertex is going to be -b over 2 a.

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And if we wanna figure out what -b over 2 a is,

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we just take the--this is the b right over here so it's a -b

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so it's -1400 over 2 a, over 2 times -30,

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which is equal to -1400 over -60.

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Negatives cancel out, we could divide the numerator and the denominator by 10.

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So this is the same thing as a 140 over 6.

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We can divide the numerator and the denominator by 3, or by, by 2.

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And you get 70, you get 70 over 3.

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And then we can just divide that, so 3 goes into 70,

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3 goes into 7 two times, 2 times 3 is 6.

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Subtract, you get a difference of 1, bring down the zero, 3 goes into 10 three times,

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3 times 3 is 9, subtract, you have, bring down, you get a 1.

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Now we're in the decimals, we bring down another zero, it becomes a 10 again,

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3 goes into 10 three times, I think you see where this is going.

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It's 23.3 repeating times. If we just keep doing this, we'll just keep getting more, more, more threes.

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Or if we just wanted to round to the nearest penny, since we're talking about selling something,

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this optimal profit, this optimal profit, will happen at a selling price of $23 and 30,

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$23 and 33 cents.

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That will optimize the yearly profit.