Maths-
General
Easy

Question

A theatre company uses the revenue function R left parenthesis x right parenthesis equals negative 50 x squared plus 250 x dollars. The cost functions of the production C left parenthesis x right parenthesis equals 450 minus 50 x. What ticket price is needed for the theatre to break even?

hintHint:

The expansion of the expression  isleft parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis text  is  end text x squared plus left parenthesis a plus b right parenthesis x plus a b.
We are asked to find the ticket price that is required to break it even for revenue and cost production function.

The correct answer is: We are asked to find the ticket price that is required to break it even for revenue and cost production function.


    Step 1 of 3 :
    The revenue function of the theatre company is R left parenthesis x right parenthesis equals negative 50 x squared minus 250 x .
    The cost functions production is C left parenthesis x right parenthesis equals 450 minus 50 x.
    To break it even, the cost function and the revenue function must be the same.
    That is, they should at least have the same value.
    That happens when they are equal, so:
    450 minus 50 x equals negative 50 x squared minus 250 x
    Step 2 of 3:
    Rearrange and simplify the equation,
    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 450 minus 50 x equals negative 50 x squared plus 250 x space space end cell row cell 50 x squared minus 250 x plus 50 x plus 450 equals 0 end cell row cell 50 x squared minus 300 x plus 450 equals 0 space space space space space space space space space space space end cell row cell x squared minus 6 x plus 9 equals 0 space space space space space space space space space space space space space space space space space space space space space space space end cell end table
    Here, we took out 50 from the equation to simplify it.
    Step 3 of 3:
    Solve the simplified equation.
    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell x squared minus 6 x plus 9 equals 0 space space space space space space space space space space space space end cell row cell x squared minus 3 x minus 3 x plus 9 equals 0 space space space space end cell row cell x left parenthesis x minus 3 right parenthesis minus 3 left parenthesis x minus 3 right parenthesis equals 0 end cell row cell left parenthesis x minus 3 right parenthesis squared equals 0 space space space space space space space space space space space space space space space space space space end cell row cell x equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space end cell end table
    Thus, they should sell the ticket at least at three dollars to even it out.

    The maximum number of solutions possible for a quadratic equation is two. It cannot have solutions more two.

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