Suppose C(x)=x^2−6x+20 represents the costs, in hundreds, to produce x thousand pens. How many pens should be produced to minimize the cost? What is the minimum cost?
Number of pens to minimize cost:
Minimum Cost:
To determine the number of pens that should be produced to minimize the cost, we need to find the vertex of the quadratic function C(x) = x^2 - 6x + 20.
The x-coordinate of the vertex can be calculated using the formula -b/(2a), where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.
For C(x) = x^2 - 6x + 20, a = 1, b = -6, and c = 20.
Using the formula, we find:
x = -(-6)/(2*1) = 6/2 = 3.
Therefore, the number of pens that should be produced to minimize the cost is 3 thousand pens.
To find the minimum cost, substitute x = 3 into the cost function C(x):
C(3) = (3)^2 - 6(3) + 20 = 9 - 18 + 20 = 11.
Hence, the minimum cost to produce the pens is 11 hundred units.
To find the number of pens that should be produced to minimize the cost, we need to find the vertex of the parabolic function represented by C(x) = x^2 - 6x + 20. The x-coordinate of the vertex gives us the number of pens to minimize cost.
The formula for the x-coordinate of the vertex can be found using the equation x = -b / (2a), where a and b are the coefficients of the quadratic equation.
In our case, the quadratic equation is C(x) = x^2 - 6x + 20, and we can see that a = 1 and b = -6.
Using the formula for the x-coordinate of the vertex, we have:
x = -(-6) / (2*1)
x = 6 / 2
x = 3
So, the number of pens that should be produced to minimize the cost is 3 thousand pens.
To find the minimum cost, we substitute the value of x back into the cost function:
C(3) = (3)^2 - 6(3) + 20
C(3) = 9 - 18 + 20
C(3) = 11
Hence, the minimum cost is 11 hundred (or $1100).
x^2 - 6 x = C-20
x^2 - 6 x + 9 = C - 11
(x-3)^2 = C-9
vertex at x = 3,000 and cost = 900