Aki’s Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by C(x) = 0.1x^2-1.2x+9.457, where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
To find the number of bicycles that minimizes the average cost per bicycle, we need to find the minimum point of the function C(x) = 0.1x^2-1.2x+9.457.
The minimum point of a quadratic function can be found using calculus. We can take the derivative of C(x) and set it equal to zero to find the critical points:
C'(x) = 0.2x - 1.2 = 0
Solving this equation for x, we get:
0.2x = 1.2
x = 1.2 / 0.2
x = 6
Therefore, the critical point is x = 6. This means that when 6 hundred bicycles are built, the average cost per bicycle will be at a minimum.
To confirm that this is indeed a minimum, we can take the second derivative of C(x):
C''(x) = 0.2
Since the second derivative is positive, this confirms that x = 6 is a minimum point.
Therefore, Aki's Bicycle Designs should build 600 bicycles to minimize the average cost per bicycle.