Question

olving problems with quadratic functions: application
photo necessities produces camera
cases. they have found that the cost,
c(x), of making x camera cases is a
quadratic function in terms of x.
the company also discovered that it
costs $15.50 to produce 3 camera cases,
$23.50 to produce 7 camera cases, and
$56 to produce 12 camera cases.
complete the function that represents the cost,
c(x), to produce x camera cases.
c(x) =
enter the correct answer.
done

Explanation:

Step1: Define quadratic cost function

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

Step2: Set up system of equations

Substitute $(3, 15.50)$, $(7, 23.50)$, $(12, 56)$:

1. $9a + 3b + c = 15.50$
2. $49a + 7b + c = 23.50$
3. $144a + 12b + c = 56$

Step3: Eliminate $c$ (eq2-eq1)

$40a + 4b = 8$ → Simplify: $10a + b = 2$ (Equation 4)

Step4: Eliminate $c$ (eq3-eq2)

$95a + 5b = 32.5$ → Simplify: $19a + b = 6.5$ (Equation 5)

Step5: Solve for $a$ (eq5-eq4)

$9a = 4.5$ → $a = \frac{4.5}{9} = 0.5$

Step6: Solve for $b$ (sub $a$ to eq4)

$10(0.5) + b = 2$ → $5 + b = 2$ → $b = -3$

Step7: Solve for $c$ (sub $a,b$ to eq1)

$9(0.5) + 3(-3) + c = 15.50$ → $4.5 - 9 + c = 15.50$ → $c = 20$

Step8: Form final cost function

Substitute $a=0.5$, $b=-3$, $c=20$ into $c(x)$.

Answer:

$c(x) = 0.5x^2 - 3x + 20$