Help on answering

Question

c(x)=150x+10000, r(x)=-2x^2+600

What is the break even point algebraically

how many dvd to produce to make money each day

How many to make to max revenue and max profit

and what is the profit function

Answer 1

(you may want to skip to near the end of this long post, where it's surmived the answer is 225 for profit maximizing output and 300 for revenue maximizing output, with 444.4 and 5.6 as breakeven output, if an "x" got left out after the 600 in the revenue function)

break even point is when cost = revenue, and profit = 0

-2x^2 + 600 = 150x +10000

2x^2 + 150x + 9,400 = 0

x^2 +75x + 4,700 = 0

the discriminant is negative, b^2-4ac<0, so there is no real solution

x=-75/2 + or - (1/2)sqr(75^2 -4(4,700))

x= -37.5 + or - .5sqr(5,625 - 18,800)

x =-37.5 + or - 57.39i

a negative imaginary level of production is the break even point

Profit = R -C = -2x^2 +600 - 150x -10000

P(x) = -2x^2 -150x - 9400

is a downward opening parabola

P = -2(x^2 +75x) -9400

P =-2(x+ 37.5)^2 -9400 +37.5^2

in vertex form:

P = -2(x+37.5)^2 -7993.75

vertex = maximum point = (-37.5, -7993.75)

maximum profit possible is negative $7,993.75 with negative production

there is no break even point. the parabola never intersects the x axis

another way to solve for maximum "profit" or minimum loss

is set Marginal Revenue = Marginal Cost

MR = R' = -4x

MC = C' = 150

x =150/-4 = -37.5

negative production makes no sense

the revenue function doesn't make sense. The more sold, the less revenue

which makes no sense

Plus it has 600 revenue for zero production

which also makes no sense, unless this is a 1930's New Deal government program to pay people not to produce and penalize them more for producing anything, to prop up market prices.

cost will always be greater than revenue

maximum revenue is when x=0 with revenue =600

odds are there are mistakes in the problem or it was copied incorrectly. It's not realistic.

just a sign change for the x^2 term would at least give a real solution

then

P(x) = 2x^2 +600 - (150x +10000)

= 2x^2 -150x -9400

breakeven point is when

2x^2 -150x -9400 = 0

x^2 -75x = 9400

(x-37.5)^2 = 9400+37.5^2

x = 37.5 + 103.95

x = 141.45

positive profits are when x>141.45

maximum profit is when x = infinity

BUT odds are an "x" got left out after the 600

R = -x^2 +600x makes sense, and is a realistic revenue function

C =150x +10,000

then breakeven point is when R=C

150x +10,000 = -x^2 +600x

x^2 -450x +10,000 = 0

x=450/2 + (1/2)sqr(450^2 -40,000)

x= 225 +.5sqr(202,500-10000)

x =225+.5sqr192,500

x =225 +438.75/2

x =225+ or - 219.4

x =5.6 DVDs to break even

also x=444.4 to break even

to make a profit 5.6<444.4

profit maximizing output = 225 DVDs

when MC=MR

150 = -2x+600

2x =450

x =225

revenue maximing output is

R'=0 = -2x+600=0

x =600/2 = 300