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^2−1.4x+8.096, where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
C(x) is just a parabola. Its minimum value is at the vertex, where
x = -b/2a = 7
To find the number of bicycles that will minimize the average cost per bicycle, we need to find the minimum value of the cost function C(x).
The average cost per bicycle is given by C(x) = 0.1x^2 - 1.4x + 8.096.
To find the minimum value, we can take the derivative of C(x) with respect to x and set it equal to zero:
C'(x) = 0.2x - 1.4 = 0
Solving this equation, we get:
0.2x = 1.4
x = 1.4 / 0.2
x = 7
Therefore, the shop should build 7 hundred bicycles to minimize the average cost per bicycle.
To minimize the average cost per bicycle, we need to find the minimum point of the function C(x).
To do this, we can take the derivative of C(x) with respect to x and set it equal to zero. The critical points obtained will help us find the minimum point.
Let's find the derivative of C(x):
C'(x) = 0.1(2x) - 1.4
Now we set C'(x) = 0 and solve for x:
0.1(2x) - 1.4 = 0
0.2x - 1.4 = 0
0.2x = 1.4
x = 1.4 / 0.2
x = 7
The critical point is x = 7.
We need to check if this is a minimum point by checking the second derivative of C(x). If the second derivative is positive, then it is a minimum point.
Let's find the second derivative of C(x):
C''(x) = 0.1(2) = 0.2
Since C''(x) = 0.2 > 0, the critical point x = 7 is a minimum point.
Therefore, to minimize the average cost per bicycle, the shop should build 7 hundred bicycles.