Solve for the breakeven point.

Question

A company has a cost equation of c=1.025x²-4x+20 and a profit equation of p=-2.125x²+6x+40. 

Answer 1

Breakeven point: c = p

1.025 x² - 4 x + 20 = 2.125 x² + 6 x + 40 Rewrite by adding the left side to the right of the equation:

1.1 x² + 10 x + 20 = 0 Rewrite with 10 as the denominator:

(11 x² + 100 x + 200)/10 = 0 Multiply both sides by 10:

11 x² + 100 x + 200 = 0 Trinomial cannot be factored, therefore use quadratic formula:

Quadratic Formula:

   

Ax² + Bx + C = 0, where  A, B and C  are coefficients, is given by :

         - B  ± √ B² - 4 AC

  x =  ————————; this equation has two solutions:

                      2 A

   - 100  ± √ 10,000 - 4*11*200 - 100 ± 100 - 8,800 - 8,800

  x =  —————————————— = ————————— = ————— = - 400

                      2*11 22 22

-8,800 9,000

x_{1 =} ————— = - 400; x_{2 =} ————— = + 409.1

22 22

Because cost is not negative, the answer is in x₂ = 409.1.

Answer 2

This looks like a trick question.

It's either a trick question, as it gives you unnecessary information about costs

or, instead of profit you might have meant revenue

to find the break even point, set profit equal to zero Profit=revenue or sales minus costs

but as written the costs are not relevant as they are already taken into account with the profit function.

I

breakeven point is when profit = 0

set the profit equation = 0 and solve for x

p=-2.125x^2 + 6x + 40 = 0

x=0 gives p=40

x=1 gives p=about 44

x=10 gives -212+100=-112

x =8 gives -136+48+40 = -48

looks like about half way between x=1 and x=8

try x=4

p=-34+24+40 =30

try x=5

p=-52.5+30+40=7..5

try x=6

p=-76.5+36+40 = -0.5

x=6 is slightly more than the break even point

x=about 5.9 probably works

x=5.9

p=-2.125(5.9)^2 +6(5.9)+40 = about -74 + 35.4+40 = 1.5

Maybe x=5.97

p = -2.125(5.97)^2 + 6(5.97)=-75.74+35.82+40 = .08

for 2 decimal places x= 5.98 or 5.99, x=5.97 is further from p=0 than 6.00

try 5.99

p=-2.125(5.99)^2 +6(5.99) + 40 = -76.25 +35.94 + 40 = -.31

try 5.98

p=-2.125(5.98)^2 + 6(5.98) + 40 = -75.99 + 35.88 + 40 = -.11

rounded to 2 decimal places output of x=5.97 comes closest to break even point with zero profits

for 3 decimal places, its about x=5.974

or you can use the quadratic formula

x=-b/2a + or - (1/2a)sqr[b^2 - 4ac]

x = -6/-5.25 + or - {sqr[36-4(-2.125)(40)]}/5.25

Answer 3

Breaking even means that the cost = profit. Therefore, we need to solve for "x" which will make both cost and profit equations equal:

1.025x² - 4x + 20 = -2.125x² + 6x + 40

Moving everything on the right side over to the left we get:

1.025x² + 2.125x² -4x -6x + 20 - 40 = 0

3.150x² -10x - 20 = 0

Graphing this on a calculator (if you have one) and finding the "zero's), we get:

x = -1.3907 and x = 4.5653

If "x" is a product, then it makes sense that it cannot be negative nor can it be a partial. Therefore, "x" would have to be 4 (4.5653 rounded down) in order for the cost to be lower than the profit.

Hope this helps!

Answer 4

1.025 x^2 - 4x + 20 = -2.125x^2 + 6x + 40

1025x^2 - 4000x + 20000 = -2125x^2 + 6000x + 40000

3150x^2 -10000x - 20000 = 0

63x^2 - 200x - 400 = 0

x = 4.565343339570432 per quadratic formula;

negative measure disqualified