Answer:
The total cost of producing 6 widgets is $231.
Step-by-step explanation:
Given : Widget wonders produces widgets. They have found that the cost, c(x), of making x widgets is a quadratic function in terms of x.
To find : The total cost of producing 6 widgets.
Solution :
Cost is given by [tex]c (x) = ax^2 + bx + d[/tex]
- Cost $23 to produce 2 widgets,
[tex]c(2) = a(2)^2 + b(2) + d[/tex]
[tex]23= 4a+2b+d[/tex] .........[1]
- Cost $103 to produce 4 widgets,
[tex]c(4) = a(4)^2 + b(4) + d[/tex]
[tex]103= 16a+4b+d[/tex] ............[2]
- Cost $631 to produce 10 widgets.,
[tex]c(10) = a(10)^2 + b(10) + d[/tex]
[tex]631= 100a+10b+d[/tex] ...........[3]
Now, we solve equation [1], [2] and [3]
Subtract equation [2]-[1] and [3]-[2]
[2]-[1] → [tex]12a+2b=80[/tex] ........[4]
[3]-[2] → [tex]84a+6b=528[/tex] .......[5]
Solving equation [4] and [5] by elimination method,
Multiply equation [4] by 3 and subtract from [5]
[tex]84a+6b-3(12a+2b)=528-3(80)[/tex]
[tex]84a+6b-36a-6b=528-240[/tex]
[tex]48a=288[/tex]
[tex]a=6[/tex]
Put in equation [4]
[tex]12(6)+2b=80[/tex]
[tex]72+2b=80[/tex]
[tex]2b=8[/tex]
[tex]b=4[/tex]
Substitute the value of a and b in [1] to get d
[tex]23= 4a+2b+d[/tex]
[tex]23= 4(6)+2(4)+d[/tex]
[tex]23= 24+8+d[/tex]
[tex]23= 32+d[/tex]
[tex]d=-9[/tex]
Substitute a=6,b=4,d=-9 in the cost equation,
The required equation form is [tex]c(x) = 6x^2 + 4x-9[/tex]
The total cost of producing 6 widgets.
Put x=6
[tex]c(6) = 6(6)^2 + 4(6)-9[/tex]
[tex]c(6) = 216+15[/tex]
[tex]c(6) =231[/tex]
Therefore, The total cost of producing 6 widgets is $231.
