Mathematics

Question

Widget Wonders produces widgets. They have found that the cost, c(x), of making x widgets is a quadratic function in terms of x.

The company also discovered that it costs $29 to produce 2 widgets, $115 to produce 4 widgets, and $757 to produce 10 widgets.

How much does it cost to make 3 widgets?

1 Answer

  • Answer:

    The total cost to make 3 widgets is $64.

    Step-by-step explanation:

    As cost is give by quadratic function such as [tex]c(x) = ax^2 + bx + d[/tex]

    As it costs $29 to produce 2 widgets. So,

    [tex]c(2) = a(2)^2 + b(2) + d[/tex]

    [tex]29= 4a+2b+d ....[A][/tex]

    As it costs $115 to produce 4 widgets. So,

    [tex]c(4) = a(4)^2 + b(4) + d[/tex]

    [tex]115= 16a+4b+d ....[B][/tex]

    As it costs $757 to produce 10 widgets. So,

    [tex]c(10) = a(10)^2 + b(10) + d[/tex]

    [tex]757= 100a+10b+d ....[C][/tex]

    In order to find the values of a, b, c and d, we have to equations [A], [B] and [C]

    Subtracting Equation [A] from [B] and [B] from [C]

     12a + 2b = 86  ........[D]

     84a + 6b = 642  ........[E]

    Multiplying Equation [D] by 3 and subtracting from [E]

    84a + 6b + 3(12a + 2b) = 642 - 86

    48a = 384

    a = 8

    Putting value of a = 8 in equation [D]

    12(8) + 2b = 86

    96 + 2b = 86

    2b = -10

    b = -5

    Substituting the value of a = 8 and b = -5 in Equation [A].

    29= 4a+2b+d

    29 = 4(8) + 2(-5) + d

    29 = 32 - 10 + d

    d = 29 + 10 - 32

    d = 7

    The required form of equation can be obtained by substituting a = 8, b = -5 and d = 7 in the cost equation. So,

    [tex]c(x) = 8x^2 - 5x + 7[/tex] is the required form of equation.

    Therefore, the total cost to make 3 widgets will be:

    Putting x = 3 in [tex]c(x) = 8x^2 - 5x + 7[/tex]

    [tex]c(3) = 8(3)^2 - 5(3) + 7[/tex]

    [tex]c(3) = 72 - 15 + 7[/tex]

    [tex]c(3) = 64[/tex]

    Hence, the total cost to make 3 widgets is $64.

    Keywords: quadratic equation, cost

    Learn more about quadratic equation from brainly.com/question/4539566

    #learnwithBrainly