Question

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.

Explanation:

Step1: Define quadratic cost function

Let $c(x)=ax^2+bx+d$, where $a$, $b$, $d$ are constants, $x$ = number of calculators, $c(x)$ = total cost.

Step2: Set up equations from given data

For $x=2$, $c(2)=45$:
$a(2)^2 + b(2) + d = 45 \implies 4a + 2b + d = 45$
For $x=4$, $c(4)=143$:
$a(4)^2 + b(4) + d = 143 \implies 16a + 4b + d = 143$
For $x=10$, $c(10)=869$:
$a(10)^2 + b(10) + d = 869 \implies 100a + 10b + d = 869$

Step3: Eliminate $d$ to simplify

Subtract first equation from second:
$(16a+4b+d)-(4a+2b+d)=143-45 \implies 12a + 2b = 98 \implies 6a + b = 49$ (Equation A)
Subtract second equation from third:
$(100a+10b+d)-(16a+4b+d)=869-143 \implies 84a + 6b = 726 \implies 14a + b = 121$ (Equation B)

Step4: Solve for $a$

Subtract Equation A from Equation B:
$(14a + b)-(6a + b)=121-49 \implies 8a = 72 \implies a=9$

Step5: Solve for $b$

Substitute $a=9$ into Equation A:
$6(9)+b=49 \implies 54 + b = 49 \implies b=49-54=-5$

Step6: Solve for $d$

Substitute $a=9$, $b=-5$ into first equation:
$4(9)+2(-5)+d=45 \implies 36-10+d=45 \implies 26+d=45 \implies d=19$

Step7: Find cost for 1 calculator

Substitute $x=1$ into $c(x)=9x^2-5x+19$:
$c(1)=9(1)^2-5(1)+19=9-5+19$

Answer:

23