the cost in dollars for a company to produce x items is give

the cost, in dollars, for a company to produce x items is given by C(x)=26+2x for x0, and the price-demand function, in dollars per item, is p(x)=30-2x for 0x15. (a) determine the profit function P(x); Since it is a quadratic function, its graph is a parabola. Does the parabola open up or down? (b) Find the vertex of the profit function P(x) using alegbra. Show algebraic work. (c) State the maxium profit and the number of items which yield that maxium profit: The Maxium profit is ________when________ items are produced and sold. (d) Determine the price to charge per item in order to maximize profit. (e) Find and Interpret the break-even points. Show alebraic work.

Cost function to produce x items C(x)=26+2x for x0

Deman function per item : p(x)=30-2x for 0x15.

Demand function to sell x iems : P(x) = xp(x) = 30x -2x^2

a) Profit = Demand - Cost = -2x^2 +30x -2x -26 = -2x^2 +28x -26

equation of parabola Opening downwars as coefficeint of x^2 is -2

b) Vertex : ax^2 +bx +c is h = -b/2a ; k = f(h)

h = -(28)/(-2*2) = 7

k = -2(7)^2 +28(7) -26 = 72

Vertex( 7,72)

c) Maximum Profit = $72 when x= 7 no. of items are produced and sold

d) price to charge per item in order to maximize profit.

plug x=7 in p(x)=30-2x

p(7) = 30 -2*7 = 30 -14 = $16

e) Break even point : cost function = demand function

30x -2x^2 = 26+2x

-2x^2 +28x -26 =0

solve the quadraticequation to find x:

we get x=1 item; = 13 items

p(1) = 30-2*1 = $28 ; p(13) = 30 -2*13 = $4

breakeven points occur at two points : ( 1, $ 28) and ( 13, $4)

the cost, in dollars, for a company to produce x items is given by C(x)=26+2x for x0, and the price-demand function, in dollars per item, is p(x)=30-2x for 0x15

the cost in dollars for a company to produce x items is give

Solution

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