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 7 calculators.

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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 7 calculators.

Mathematics
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Quadratic function is \[C(x) = ax^2+bx+c\] Given points are C(2) = 45, C(4) = 143, and C(10) = 869. Plug them in for C and x to get a system of equations to solve for the constants a, b, and c.
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Your points follow the pattern (number of calculator, cost). $45 to produce 2 calculators means the point is (2, 45). Plug that into the C(x) equation above. \[C(x) = ax^2+bx+c\] \[45=a(2)^2+2b+c\] \[45=4a+2b+c\] You need to do this with the other two data points to get two more equations

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You need three equations to solve to find the values of the coefficient of x squared (a), the coefficient of x (b), and the numerical term (c) in the quadratic equation that models the cost, c(x). The three equations are: 45 = 4a + 2b + c ..............(1) 143 = 16a + 4b + c ..........(2) 869 = 100a + 10b + c.......(3) Subtracting (2) from (1) eliminates c and gives us: 98 = 12a + 2b ...................(4) And subtracting (2) from (3) gives us: 726 = 84a + 6b = 726 .......(5) If we multiply (4) by 3 we get: 294 = 36a + 6b .................(6) And subtracting (6) from (5) eliminates the terms in b and gives us: 432 = 48a ..........................(7) which enables us to find the value of a to be 9. Plugging the value of a into (6) enables us to find the value of b. The values of a and b can then be plugged into (1) and the value of c can be found. Finally you can construct the quadratic that models c(x) and find the total cost of producing 7 calculators.

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