quick computing company produces calculators. they have found that the cost, c(x), of making x calculators…

quick computing company produces calculators. they have found that the cost, c(x), of making x calculators is a quadratic function in terms of x. the company also discovered that it costs $61 to produce 3 calculators, $165 to produce 7 calculators, and $385 to produce 12 calculators. find the total cost of producing 6 calculators. enter the correct answer.

quick computing company produces calculators. they have found that the cost, c(x), of making x calculators is a quadratic function in terms of x. the company also discovered that it costs $61 to produce 3 calculators, $165 to produce 7 calculators, and $385 to produce 12 calculators. find the total cost of producing 6 calculators. enter the correct answer.

Answer

Explanation:

Step1: Set up quadratic equation

Let $c(x)=ax^{2}+bx + c$. We know that $c(3)=9a + 3b + c=61$, $c(7)=49a+7b + c = 165$, $c(12)=144a+12b + c=385$.

Step2: Subtract equations

Subtract the first - equation from the second: $(49a + 7b + c)-(9a + 3b + c)=165 - 61$, which simplifies to $40a+4b = 104$, or $10a + b=26$ (Equation 1). Subtract the second equation from the third: $(144a+12b + c)-(49a + 7b + c)=385 - 165$, which simplifies to $95a+5b = 220$, or $19a + b = 44$ (Equation 2).

Step3: Solve for a

Subtract Equation 1 from Equation 2: $(19a + b)-(10a + b)=44 - 26$, so $9a=18$, and $a = 2$.

Step4: Solve for b

Substitute $a = 2$ into Equation 1: $10\times2 + b=26$, so $b=6$.

Step5: Solve for c

Substitute $a = 2$ and $b = 6$ into $9a + 3b + c=61$. We get $9\times2+3\times6 + c=61$, $18 + 18 + c=61$, $c=25$.

Step6: Find $c(6)$

So $c(x)=2x^{2}+6x + 25$. Then $c(6)=2\times6^{2}+6\times6 + 25=2\times36+36 + 25=72+36 + 25 = 133$.

Answer:

133