Question 258477: aki's bicycle designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by C(x)=0.5x^2-1.2x+2.725, how many bikes should the shop build to minimize the average cost per bike? thank you
Answer by mathbath(13)   (Show Source):
You can put this solution on YOUR website! The given function to determine the cost per bicycle is ..
C(x)=0.5x^2-1.2x+2.725 .
The standard form of a quadratic equation is ..
f(x)=ax^2+bx+c
Comparing the above two equations we can say that the Cost function C(x) is in a standard quadratic equation form where..
a = 0.5, b = -1.2, c = 2.725
Now if any quadratic equation is plotted on a graph we get a parabola (which looks like the letter "U" or an inverted "U")
For quadratic equations..
if a > 0 , The parabola is in the form of a letter "U" i.e., it opens upwards. This also means that the function has a minimum which is at the vertex of the parabola.
Considering the standard form of quadratic equation, the vertex of the parabola has co-ordinates {-b/2a , f(-b/2a)}
So,
x = -b/2a
Substituting the values for a and b we can find the value of x.
x = -(-1.2)/2(0.5) = 1.2/1 = 1.2
So, the Cost function has a minimum when x = 1.2.
Since the Cost function is for 100x bicycles , aki's bicycles needs to manufacture atleast 100(1.2) bicycles to minimize the cost per bicycle.
So the total no. of required bicycles is 120
Verify:
Now lets verify our answer.
The value of x that we got is 1.2. Lets substitute this in the Cost function C(x)
C(x) = 0.5(1.2)^2 - 1.2(1.2) + 2.725
= 0.5(1.44) - 1.44 + 2.725
= 0.72 + 1.285
= 2.005
Lets raise the value of x to 2 and substitute in the cost function C(x)
C(x) = 0.5(2)^2 - 1.2(2) + 2.725
= 0.5(4) - 2.4 + 2.725
= 2 + 0.325
= 2.325
So as you can see as we raised the value of x from 1.2 to 2 the C(x) function jumped up from 2.005 to 2.325. In other words what this means is the value of x=1.2 is the minimum for this function.
Hope this helps.
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