Question 394690: Aki's Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by  , where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
The shop should build ___ bicycles.
Found 3 solutions by nerdybill, jrfrunner, ewatrrr:
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! Aki's Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by , where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
.
You simply need to find the vertex of:
this occurs when:
x = -b/(2a)
x = -(-0.4)/(2*0.1)
x = (0.4)/(0.2)
x = 2
.
solution: 200 bicycles
Answer by jrfrunner(365)  (Show Source):
You can put this solution on YOUR website! Avg Cost/bike =
--
This is a parabola with its "a" coefficient 0.1 being positive so the curve opens upward indicating that the vertex is a minimum. vertex is located at x=-b/(2a)= -(-0.4)/(2*0.1)=2, so the shop should build 200 bikes to minimize average cost
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Another way to do this is to take the first derivative of the avg cost function
 this gives the instantenous slope and we want to find the extremas which occur when the instantenous slope=0 (also known as critical points)
0.2x-0.4=0
x=2 or 200 bikes since x is in hundreds of bikes.
to determine if this is a minimum or maximum extrema, take the second derivative
C"(x)=0.2 since this is positive at the critical point (and all points in this case) this means that it curves upward and the critical point is a minimum
Answer by ewatrrr(24785)  (Show Source):
You can put this solution on YOUR website!
Hi
C(x)=0.1x^2-0.4x+ 7.898 |C(x)average cost per bicycle, building x-hundred
C(x)=0.1(x^2- 4x +7.898
C(x)=0.1(x-2)^2 -.4 + 7.898
C(x)=0.1(x-2)^2 + 7.498 vertex is Pt(2,7.498)
200 bicycles should be built to minimize the average cost per bicycle
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