Completing the square math help?

Price is x.

Sales per week = 30 - (x - 20) = (50 - x)

Material costs per week = 10(50 - x)

Takings per week = x(50 - x)

Profit = x(50 - x) - 10(50 - x) = -x^2 + 60x - 500

To make it easier, (- Profit) = x^2 -60x + 500

= x^2 -60x + (30)^2 - 400 (this is completing the square) = (x - 30)^2 - 400

You want (- profit) to be as small as possible. The bracket is as small as possible (= 0) when x = 30, making the (- profit) = -400.

So at a price of $30 her profit is maximised at $400 per week.

Profit = price x portraits. Her price starts at 20 and increases by x. The number of portraits is currently 30 per week, and will decrease by x. So:

P = (20 + x)(30 - x) or multiplied out:

P = -x^2 + 10x + 600

This is a quadratic, hence graphs into a parabola. And since the 'a' value is negative, it's a 'frowny' parabola. To figure out what x-value would maximize the profit, I would just use the formula for the axis of symmetry: x = -b/2a (with the equation set up as ax^2 + bx + c).

You mention 'completion the square', but that's what one usually does to find x-intercepts, not maximums. The x-intercepts would be the points where the profit is exactly zero, and therefore wouldn't be as useful.

Hope this helps.

@Steve the Freshman - I was operating under the assumption that this was an algebra question, not calculus.

Profit P, @ $20 = 30(20 -10)

After some price increase $x, Profit (Px) = (30 - x)(20 - 10 + x) = (30 - x)(10 + x)

P(x) = 300 + 20x - x^2 like all quadratics, is a parabola

Because the p^2 term is negative, the parabola opens downwards and its maximum is the vertex, the x value of which for ax^2 + bx + c is given by -b/2a

In this instance the vertex is at x = -20/-2 = 10

Her new price should be $30, which will net her $400 in profit

See:http://www.mathwarehouse.com/geometry/parabola/sta...

let us assume that she increases the price by $x

so profit = (20+x)*(30-x) = -x² +10x - 600

for maximizing it, differentiate p wrt x and equate that to zero

dp/dx = -2x +10 = 0

so x = 5

p(5) = (20+5)*(30-5) = 625

@BailiffQuimby

"@Steve the Freshman - I was operating under the assumption that this was an algebra question, not calculus"

I really don't know why you said that. I was operating under the assumption that it was calculus, so I don't need your comment.

how much profit she want?