SOLUTION: C(x)= 0.1x^2-0.7x+2.425, where C(x) is hundreds of dollars. How many bicycles should be built in order to minimize the average cost per bicycle

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Question 1204231: C(x)= 0.1x^2-0.7x+2.425, where C(x) is hundreds of dollars. How many bicycles should be built in order to minimize the average cost per bicycle
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Based on this previous problem here
https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1177138.html
It appears there should be the instruction that "x hundred bicycles are built"

x = number of bikes in hundreds
eg: x = 3 means 300 bikes are made
C(x) = 0.1x^2-0.7x+2.425 is the cost in hundreds of dollars.

%28C%28x%29%29%2Fx+=+%280.1x%5E2-0.7x%2B2.425%29%2Fx represents the average cost, in hundreds of dollars, of making x hundred bikes.

Use calculus or a graphing calculator to determine the local min of %28C%28x%29%29%2Fx
GeoGebra and Desmos are graphing tools I recommend.
If you have a TI83 or TI84, then that works as well.

When x > 0, the lowest point happens at roughly (4.92443, 0.28489)

x = 4.92443 leads to 100*4.92443 = 492.443

Of course it's not possible to build 0.443 of a bike.
We look at integer values to the left and right of 492.443
We'll look at integer values 492 and 493 which correspond to x = 4.92 and x = 4.93 respectively.

If x = 4.92, then the average cost is:
%28C%28x%29%29%2Fx+=+%280.1x%5E2-0.7x%2B2.425%29%2Fx

%28C%284.92%29%29%2F%284.92%29+=+%280.1%284.92%29%5E2-0.7%284.92%29%2B2.425%29%2F%284.92%29

%28C%284.92%29%29%2F%284.92%29+=+0.28488617886179 approximately

%28C%284.92%29%29%2F%284.92%29+=+0.2849 approximately

Since %28C%28x%29%29%2Fx is the average cost in hundreds of dollars, it means the result 0.2849 represents 0.2849*100 = 28.49 dollars.
It costs around 28.49 dollars per bike, on average, if 492 bikes are made.

Now try x = 4.93
%28C%28x%29%29%2Fx+=+%280.1x%5E2-0.7x%2B2.425%29%2Fx

%28C%284.93%29%29%2F%284.93%29+=+%280.1%284.93%29%5E2-0.7%284.93%29%2B2.425%29%2F%284.93%29

%28C%284.93%29%29%2F%284.93%29+=+0.2848864097363 approximately

%28C%284.93%29%29%2F%284.93%29+=+0.2849 approximately
We get the same average cost as before.

It is up to the company to decide whether to make 492 bikes or 493 bikes.
Either option produces the lowest average cost of $28.49 dollars per bike.