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 $45 to produce 2 calculators, $143 to produce 4 calculators, and $869 to produce 10 calculators. find the total cost of producing 1 calculators. enter the correct answer.

Answer

Explanation:

Step1: Set up the quadratic - function equation

Let $c(x)=ax^{2}+bx + c$. We know that $c(2)=4a + 2b + c=45$, $c(4)=16a+4b + c = 143$, and $c(10)=100a + 10b + c=869$.

Step2: Subtract equations to eliminate $c$

Subtract the first equation from the second: $(16a + 4b + c)-(4a + 2b + c)=143 - 45$, which simplifies to $12a+2b = 98$ (divide by 2) $\Rightarrow6a + b=49$. Subtract the second equation from the third: $(100a+10b + c)-(16a + 4b + c)=869 - 143$, which simplifies to $84a+6b = 726$ (divide by 6) $\Rightarrow14a + b=121$.

Step3: Solve the system of linear equations for $a$ and $b$

Subtract the equation $6a + b=49$ from $14a + b=121$: $(14a + b)-(6a + b)=121 - 49$, $8a=72$, so $a = 9$. Substitute $a = 9$ into $6a + b=49$, we get $6\times9 + b=49$, $54 + b=49$, $b=-5$.

Step4: Find $c$

Substitute $a = 9$ and $b=-5$ into $4a + 2b + c=45$, $4\times9+2\times(-5)+c=45$, $36-10 + c=45$, $26 + c=45$, $c = 19$.

Step5: Find $c(1)$

Substitute $x = 1$ into $c(x)=9x^{2}-5x + 19$, $c(1)=9\times1^{2}-5\times1 + 19=9 - 5+19=23$.