Answer:
Explanation:
a. To minimize the average cost, we need to find the value of x that minimizes the cost function C(x) = x² - 40x + 500.
This is the same as finding the minimum value of the parabola represented by the function. The vertex of the parabola is the point at which the parabola reaches its minimum or maximum value. The vertex form of a parabola is given by C(x) = a(x - h)² + k, where (h, k) is the vertex.
By comparing the given function with the vertex form of a parabola, we can find the vertex:
C(x) = x² - 40x + 500
= a(x - h)² + k
= (x - h)² + k
The x-coordinate of the vertex h = 40/2 = 20
So the minimum average cost occurs when 20 televisions are made per week.
b. To find the minimum average cost, we need to substitute the value of x = 20 into the cost function:
C(x) = x² - 40x + 500
= 20² - 40*20 + 500
= 400 - 800 + 500
= 100
So the minimum average cost is $100 per television.
If the average cost is $200 per television, we can set C(x) = 200 and solve for x:
200 = x² - 40x + 500
0 = x² - 40x + 300
This is a quadratic equation that can be solved by factoring or by using the quadratic formula.
we can factor it as
0 = (x-50)(x-6)
x = 50 or x = 6
The number of televisions made per week if the average cost is $200 per television is 50 or 6. Since 50 is much higher than 20, the minimum average cost, it is not physically possible to make 50 televisions with the average cost of $200 per television. So the answer is 6 televisions per week.