Question

an aircraft factory manufactures airplane engines. The unit cost c (the cost in dollars to make each airplane engine) depends on the number of engines made. If x engines are made, then the unit cost is given by the function c(x)=0.3x2- 96x+26,919. How many engines must be made to minimize the unit cost?

Answer 1

The vertex is the minimum or maximum point of a parabola.

The formula to find the vertex is, where a, b and c are values in the standard form of quadratic equation is:

x min/max = - b / 2a

If a > 0, then the parabola the y-value of the vertex is the minimum value of the function.

If a < 0, then the parabola the y-value of the vertex is the maximum value of the function.

The formula to find the vertex is, where a, b and c are values in the standard form of quadratic equation is:

x min/max = - b / 2a

If a > 0, then the parabola the y-value of the vertex is the minimum value of the function.

If a < 0, then the parabola the y-value of the vertex is the maximum value of the function.

In your case:

c(x) = 0.3 x² - 96 x + 26 919

a = 0.3 , b = - 93 , c = 26 919

a > 0 , so the y-value of the vertex is the minimum value of the function.

x min = - b / 2a = - ( - 96 ) / 2 ∙ 0.3 = 96 / 0.6 = 160

160 engines must be made to minimize the unit cost.

c min = c ( 160 ) = 0.3 ∙ 160² - 96 ∙ 160 + 26 919 = 19 239

Answer 2

My typo.

b is not - 93

b = - 96

a = 0.3 , b = - 96 , c = 26 919