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.
complete the function that represents the cost, c(x), to produce x widgets.
c(x) =
enter the correct answer.

Explanation:

Step1: Define quadratic cost function

Let $c(x) = ax^2 + bx + c$, where $a$, $b$, $c$ are constants.

Step2: Set up equations from given data

For $x=2$, $c(2)=29$:
$a(2)^2 + b(2) + c = 29 \implies 4a + 2b + c = 29$
For $x=4$, $c(4)=115$:
$a(4)^2 + b(4) + c = 115 \implies 16a + 4b + c = 115$
For $x=10$, $c(10)=757$:
$a(10)^2 + b(10) + c = 757 \implies 100a + 10b + c = 757$

Step3: Eliminate c to simplify equations

Subtract first equation from second:
$(16a + 4b + c) - (4a + 2b + c) = 115 - 29$
$12a + 2b = 86 \implies 6a + b = 43$ (Equation 4)
Subtract second equation from third:
$(100a + 10b + c) - (16a + 4b + c) = 757 - 115$
$84a + 6b = 642 \implies 14a + b = 107$ (Equation 5)

Step4: Solve for a

Subtract Equation 4 from Equation 5:
$(14a + b) - (6a + b) = 107 - 43$
$8a = 64 \implies a = 8$

Step5: Solve for b

Substitute $a=8$ into Equation 4:
$6(8) + b = 43 \implies 48 + b = 43 \implies b = -5$

Step6: Solve for c

Substitute $a=8$, $b=-5$ into first equation:
$4(8) + 2(-5) + c = 29 \implies 32 - 10 + c = 29 \implies 22 + c = 29 \implies c = 7$

Step7: Form the cost function

Substitute $a=8$, $b=-5$, $c=7$ into $c(x)=ax^2+bx+c$.

Answer:

$c(x) = 8x^2 - 5x + 7$