Aki's Bicycle designs has determined that when x hundred bicyles are built, the average cost per bicycle is given by C(x)=0.6^2-0.5x+4.021, where C(x) is in hundreds of dollars. How many biccles should the shop build to minimize the average cost per bicycle?
The shop should build _____ bicycles.
Could you please help me!!!!
Don't you mean
C(x)=0.6 x^2 - 0.5x + 4.021 ?
I think you left out an x.
Since you are not taking calculus, I will assume that you are supposed to answer this question by turning C(x) into a "perfect square" plus a constant.
C(x)= 0.6 x^2 - 0.5x + 4.021
0.6 [x^2 - (5/6)x + (5/12)^2] + 3.916
= 0.6(x - 5/12)^2 + 3.916
This is a minimum when x = 5/12. that means the number of bicyles is (5/12)*100 = 41.7
Pick x = 0.42 instead of 0.41, since it is closer to 5/12 and 100x must be an integer.
Proof: If x = 0.41, C(x) = 3.91686
if x = 0.42, C(x) = 3.91684
if x = 0.43, C(x) = 3.91694
To minimize the average cost per bicycle, we need to find the minimum value of the function C(x) = 0.6x^2 - 0.5x + 4.021.
To do so, we can use calculus by finding the derivative of C(x) and setting it equal to zero. This will give us the critical point where the minimum occurs.
1. Differentiate C(x) with respect to x:
C'(x) = 2(0.6)x - 0.5
2. Set C'(x) = 0 and solve for x:
2(0.6)x - 0.5 = 0
1.2x - 0.5 = 0
1.2x = 0.5
x = 0.5 / 1.2
x = 0.4167
3. Since the problem states that x represents hundreds of bicycles, we need to round the result to the nearest whole number:
x ≈ 0.42 (rounded to two decimal places)
Therefore, the shop should build approximately 42 bicycles to minimize the average cost per bicycle.
To find the number of bicycles that the shop should build to minimize the average cost per bicycle, we need to find the value of x that minimizes the function C(x).
The function given is C(x) = 0.6^2 - 0.5x + 4.021, where C(x) is in hundreds of dollars.
To minimize this function, we need to determine the critical points by taking the derivative of C(x) with respect to x and setting it equal to zero.
C'(x) = -0.5
Setting C'(x) = 0, we get:
-0.5 = 0
This equation has no solution because -0.5 is a constant and cannot be equal to zero.
Since the first derivative test cannot be used to find the minimum, we need to check the behavior of the function at the endpoints of the possible range.
The number of bicycles cannot be negative, so we will consider the number of bicycles starting from x = 0.
Let's plug in some values of x to see how the average cost per bicycle changes.
When x = 0, C(x) = (0.6^2) + 4.021 = 4.021 (in hundreds of dollars).
When x = 1, C(x) = (0.6^2) - 0.5(1) + 4.021 = 4.051 (in hundreds of dollars).
When x = 2, C(x) = (0.6^2) - 0.5(2) + 4.021 = 4.021 (in hundreds of dollars).
From these values, we see that the average cost per bicycle is at its minimum when x = 2.
Therefore, the shop should build 2 hundred bicycles to minimize the average cost per bicycle.