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 -0.5x+ 9.295. Where C(x) is in hundreds of dollars, how many bicycles should the shop build to minimize the average cost per bicycle?
Thank you for your help
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Aki's Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by C(x)equals
0.1xsquaredminus1.1xplus4.514
,
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 will minimize the average cost per bicycle, we need to find the minimum point of the cost function, c(x).
Step 1: Identify the function
The function given is c(x) = 0.1x^2 - 0.5x + 9.295, where C(x) is in hundreds of dollars.
Step 2: Find the derivative
To find the minimum point, we need to find the critical points. For this, we take the derivative of the cost function, c'(x), with respect to x.
c'(x) = d/dx (0.1x^2 - 0.5x + 9.295)
= 0.2x - 0.5
Step 3: Set the derivative equal to zero
To find the critical points, we set c'(x) = 0 and solve for x:
0.2x - 0.5 = 0
0.2x = 0.5
x = 0.5 / 0.2
x = 2.5
Step 4: Determine the minimum point
Now that we have the critical point x = 2.5, we need to confirm that it is a minimum. We can do this by evaluating the second derivative of the cost function, c''(x).
c''(x) = d^2/dx^2 (0.1x^2 - 0.5x + 9.295)
= 0.2
Since the second derivative is positive (0.2), this indicates a minimum point at x = 2.5.
Step 5: Determine the number of bicycles
Since x represents the number of hundred bicycles, the number of bicycles Aki's Bicycle Designs should build to minimize the average cost per bicycle is given by:
x = 2.5 * 100
x = 250
Therefore, the shop should build 250 bicycles to minimize the average cost per bicycle.
if you graph and find the minimum, you would find the point (2.5, 8.67)
This means that they should sell 250 bikes at the price of $8.67