The cost to produce bottled spring water is given by C(x)=16...

The cost to produce bottled spring water is given by $C(x)=16 x-63,$ where $x$ is the number of thousands of bottles. The total income (revenue) from the sale of these bottles is given by the function $R(x)=-x^{2}+326 x-7463$. since profit $=$ revenue $-$ cost, the profit function must be $P(x)=-x^{2}+310 x-7400$ (verify). how many bottles sold will produce the maximum profit? What is the maximum profit?

Transcript

00:01 So, we're given an expression to represent the cost of selling bottled spring water, and we're also given an expression for the revenue, and we need to first verify that this is the profit expression.

00:15 So if we think about profit, profits is equal to revenue minus cost, so the amount of money you bring in minus the amount of money you spend.

00:24 So if we think about what we're given, we know that our revenue is negative x squared plus 326x minus 7 ,463, and we're going to subtract from that our cost, which is 16x minus 36.

00:40 So to verify that this is actually the profit function, we're going to combine like terms.

00:46 So here's my x squared, that's the only x squared, so i'm going to write x squared there.

00:50 I have 326x, and i'm going to subtract, remember, subtract, 16x.

00:56 So that will give us 310x.

01:00 And then the last thing we're going to do is take our negative 7 ,463, and we're going to subtract negative 63.

01:08 So we're subtracting a negative, which means we're adding, which gives us negative 7 ,400.

01:15 So if we look at what we were given, this verifies that that is indeed the profit function.

01:21 Now what we're going to do is find the maximum, so to do that i'm going to clean up some stuff, and then we're going to make some room to find our maximum profit, and how many bottles will make that maximum profit.

01:32 So in order to find the maximum, we're going to complete the square so we can find the vertex of this parabola.

01:38 So first thing we're going to do is move the non -squared variable term over to the other side.

01:44 So i end up with p of x plus 7 ,400 equals negative x squared plus 310x.

01:53 Now what we're going to do is we're going to factor out that negative, so we end up with negative x squared minus 310x.

02:00 And we're going to complete the square to find our maximum, so i'm going to go ahead and write a plus blank, and that's where we're going to add in a number to keep our equation balanced.

02:09 So on the other side we still have p of x plus 7 ,400.

02:15 So now to do our complete the square, we're going to rewrite this side here as a perfect square...

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