What is the start up cost? And How many must they sell to break even?

Question

Harry and his sister started selling books. The profit function, in dollars for selling x books is        f(x)=-0.5x^2+25x-112.5 

Answer 1

The start-up cost (negative profit) is the cost at time=0 (sold x=0 books):

      f(x)=-0.5x^2+25x-112.5

      f(0) =   0      + 0   -112.5

     $112.50

 

The break-even point (point where profits = start-up costs) is where f(x)=0:

       f(x)=-0.5x^2+25x-112.5 

        0 = -0.5x^2+25x-112.5 

        0.5x^2 - 25x + 112.5 = 0                    (rearrange)

        x^2 - 50x + 225 = 0                       (multiply by 2 to get rid of fractions)

       (x-5)(x-45)=0                    (either factor or use quadratic formula)

             Costs are higher than profits when selling less than 5 books (break-even point) and again when selling more than 45 books (so, don't do that!).

Answer 2

 

Hi James,

 

When strting up, i.e., x = 0, f(0) = -112.5

 

So, the start-up cost is 112.5 dollars.

 

If we solve f(x) = 0, we get x = 5 or x = 45.

 

So, they need to sell 5 books to break even.