The daily production costs C (in dollars per unit) for a manufacturer of lighting fixtures are given by the quadratic function C(X)=800-10X+0.25X2 where x is the number of units produced. How many fixtures should be produced to yield a minimum cost per unit?
the minimum cost is found at the vertex of the parabola. As you recall, that is at x = -b/2a, or in this case
x = 10/.5 = 20
To find the number of fixtures that should be produced to yield a minimum cost per unit, we need to determine the value of x that corresponds to the vertex of the quadratic function.
The quadratic function is in the form: C(X) = 800 - 10X + 0.25X^2
The vertex form of a quadratic function is: C(X) = a(X - h)^2 + k, where (h, k) represents the vertex.
By comparing the two forms, we can determine that h = -b/2a , where a = 0.25 and b = -10.
h = -(-10) / (2 * 0.25)
= 10 / 0.5
= 20
So, the x-coordinate of the vertex is 20.
Therefore, the manufacturer should produce 20 lighting fixtures to yield a minimum cost per unit.
To determine the number of fixtures that should be produced to yield a minimum cost per unit, we need to find the vertex of the quadratic function. The vertex represents the lowest point (minimum) on the graph of the function.
The quadratic function C(X) = 800 - 10X + 0.25X^2 is in the form of a quadratic equation, ax^2 + bx + c.
In this case, a = 0.25, b = -10, and c = 800.
The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
Plugging in the values, we get:
x = -(-10) / (2 * 0.25)
x = 10 / 0.5
x = 20
So, the x-coordinate of the vertex is 20, which represents the number of fixtures that should be produced to yield a minimum cost per unit.
Thus, the manufacturer should produce 20 fixtures to minimize the cost per unit.