Aki's Bicycle Design has determined that when x hundred bicycles are built, the average cost per bicycle =0.9x^2-1.7x+10.861, when C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
C(x)=0.9x^2-1.7x+10.861 =
C(x)=0.9(x^2-1.889x)+10.861=
My question is how do they come up with the 1.889 in the equation?
For Further Reading
* algebra - drwls, Thursday, March 6, 2008 at 4:44am
They are using the method of "completing the square" and may be trying to get a perfect square inside the parentheses.
-0.9 x 1.88888... and -1.7 are the same thing. They rounded off the the 1.88888.. to 1.889.
With a complete square inside parentheses, C(x) becomes
0.9(x^2 - 1.889x + 0.8920) + 10.861 - 0.8028
= 0.9(x- 0.9445)^2 + 10.058
You get minimum average cost when x = .9445
Okay, that makes sense, but now where do you get the 0.8920 in the equation from?
If you understand the subject of completing the square, you should know that the constant term has to be the square of half of the coefficient of x,
(1.889/2)^2. So, that constant had to be added inside the parentheses and subtracted somewhere else (after being multipled by 0.9)
Thank you very much. I am not very good at Algebra, but maintaining an "A". I want to totally understand how the numbers are arrived at and then it makes complete sense.
You are doing the right thing. Once you understand the logic of each step, it becomes easier to do them yourself. Otherwise, it's like watching rabbits pulled out of a hat in a magic show.
To find the constant term inside the square parentheses, you need to take half of the coefficient of x (in this case, -1.889) and square it.
So, (-1.889 / 2)^2 = 0.8920.
This is done to complete the square and rewrite the expression in the form (x - h)^2, where h is a constant value. In this particular case, we have (x - 0.9445)^2.
After completing the square, you subtract the constant outside the square parentheses (0.8028 in this case) to maintain the equality.
Therefore, the final expression becomes 0.9(x - 0.9445)^2 + 10.058.
To minimize the average cost per bicycle, you need to find the value of x that minimizes this expression. In this case, they found that x = 0.9445 results in the minimum average cost.