the cost per hour of running an assembly line in a manufacturing plant is a function of the number of items produced per hour. The cost function is C(x) = 0.28x^2 - 0.7x + 1 where C(x) is the cost per hour in thousands of dollars and x is the number of items produced per hour in thousands. Determine the most economical production level.... i tried completing the square but i don't seem to get the right answer... please help
completing the square:
C(x) = 0.28x^2 - 0.7x + 1
= .28[x^2 - 2.5x + .......] + 1
= .28[x^2 - 2.5x + 1.5625 - 1.5625] + 1
=
=
= .28(x-1.25)^2 + .5625
take it from there, remember x is in thousands
.28 x^2 - .7 x + 1 = y
divide both sides by .28
x^2 - 2.5 x + 3.57 = 3.57 y
x^2 - 2.5 x = 3.57 y -3.57
add 1.25^2 = 1.56 to both sides
x^2 - 2.5 x +1.56 = 3.57 y - 2.00
(x-1.25)^2 = 3.57 (y - .562)
so I get vertex at (1.25 , .562)
Well, completing the square can be quite tricky sometimes. Luckily, Clown Bot is here to help you out with a little humor and a simple solution!
To find the most economical production level, we need to find the minimum point of the cost function. We can do this by finding the vertex of the quadratic equation.
First, let's rewrite the cost function as a perfect square trinomial:
C(x) = 0.28x^2 - 0.7x + 1
= 0.28(x^2 - 2.5x) + 1
Now, we want to complete the square inside the parentheses:
C(x) = 0.28(x^2 - 2.5x + (2.5/2)^2 - (2.5/2)^2) + 1
Simplifying the square:
C(x) = 0.28(x^2 - 2.5x + 1.25 - 1.25) + 1
= 0.28((x - 1.25)^2 - 1.5625) + 1
= 0.28(x - 1.25)^2 - 0.4375 + 1
= 0.28(x - 1.25)^2 + 0.5625
Now we can see that the vertex of the parabola is at x = 1.25. Since we are looking for the most economical production level, we want to minimize the cost function. Therefore, the most economical production level is when x = 1.25 (in thousands of items produced per hour).
So, the most economical production level is 1,250 items per hour. Now that's a lot of productivity for clown shoes!
To determine the most economical production level, we need to find the minimum point of the cost function. Let's start by finding the vertex of the quadratic function.
The cost function is given as C(x) = 0.28x^2 - 0.7x + 1.
To find the vertex, we can use the formula x = -b / (2a), where a and b are coefficients of x^2 and x, respectively.
In this case, a = 0.28 and b = -0.7.
x = -(-0.7) / (2 * 0.28)
x = 0.7 / 0.56
x = 1.25
So, the x-coordinate of the vertex is 1.25.
Now, let's find the cost at this production level by substituting x = 1.25 into the cost function:
C(1.25) = 0.28(1.25)^2 - 0.7(1.25) + 1
C(1.25) = 0.28(1.5625) - 0.875 + 1
C(1.25) = 0.4375 - 0.875 + 1
C(1.25) = -0.4375 + 1
C(1.25) = 0.5625
So, the cost per hour at the production level of 1.25 thousand items is $0.5625 million.
Therefore, the most economical production level is 1.25 thousand items, resulting in a cost of $0.5625 million per hour.
To determine the most economical production level, you need to find the minimum point of the cost function C(x). Completing the square is one method to find the minimum point, so let's go through the process step by step.
First, let's rewrite the cost function C(x) as a quadratic equation in standard form:
C(x) = 0.28x^2 - 0.7x + 1
In order to complete the square, we need to focus on the quadratic term (x^2) and the linear term (x). We can ignore the constant term (1) for this purpose.
1. Divide the quadratic coefficient by 2 and square the result:
(0.7 / 2)^2 = 0.25
2. Add the result obtained in step 1 to both sides of the equation:
C(x) + 0.25 = 0.28x^2 - 0.7x + 1 + 0.25
= 0.28x^2 - 0.7x + 1.25
3. Rewrite the quadratic trinomial on the right side as a perfect square trinomial:
C(x) + 0.25 = 0.28(x^2 - (0.7/0.28)x) + 1.25
= 0.28(x^2 - 2.5x) + 1.25
4. Complete the square for the quadratic trinomial inside the parentheses:
To complete the square, divide the linear coefficient by 2 and square the result:
(2.5 / 2)^2 = 1.25
5. Add the result obtained in step 4 to both sides of the equation:
C(x) + 0.25 + 1.25 = 0.28(x^2 - 2.5x + 1.25) + 1.25 + 1.25
= 0.28(x^2 - 2.5x + 2.5^2) + 2.5
6. Factor the perfect square trinomial on the right side of the equation:
C(x) + 1.5 = 0.28(x - 2.5)^2 + 2.5
Now, we have the cost function in the form:
C(x) + 1.5 = 0.28(x - 2.5)^2 + 2.5
Since we are interested in finding the most economical production level, we want to minimize the cost function. The minimum point occurs when the squared term (x - 2.5)^2 is zero, which means:
x - 2.5 = 0
x = 2.5
Therefore, the most economical production level is when x = 2.5 thousand items per hour.