The cost to produce a product is modeled by the function f(x) = 3x2 − 6x + 11 where x is the number of products produced. complete the square to determine the minimum cost of producing this product.

Because this is a positive parabola that opens upwards, it natrually has a minimum value for the vertex. To complete the square we will first set the quadratic equal to 0 then move the constant over by subtraction. That gives us . One very important rule to follow about completing the square is that the leading coefficient ALWAYS be a +1. Our is a 3. So we will factor it out like this: . Next step is to take half the linear term, square it, and add it to both sides. Our linear term is 2. Half of 2 is 1, and 1 squared is 1. So we add 1 to both sides. BUT we cannot forget about that 3 we factored out. It refuses to be ignored. It is a multiplier. So when we add 1 into the set of parenthesis, we are REALLY adding in 3*1 = 3. We add 3 to the right side. All that looks like this: . The right side simplifies to -8, easy enough. What we did this for, this whole entire process, was to create a perfect square binomial on the left that will serve as the x coordinate of the vertex (actually the "h" since the vertex coordinates are (h, k)). Setting up that binomial looks like this now: . We move the -8 back over by addition, and find that the coordinates for the vertex are (1, 8). If x (h) is 1, that is "1 product" since x is products. If y (k) is 8, that is $8. What our vertex tells us in this particular situation is that it costs $8 to make 1 item.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240