The annual inventory cost C for a manufacturer is C = 1,008,00/Q + 6.3Q where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 350 to 351, and compare this with the instantaneous rate of change
when Q = 350.
differential
∆C/∆Q = C(351)-C(350) = (1008000/351 + 6.3*351)-(1008000/350 + 6.3*350) = -1.90512
dC/dq = 6.3 - 1008000/Q^2 = -1.92857
Well, let's plug in the values and see what happens.
When Q is 350, the annual inventory cost is:
C = 1,008,000/350 + 6.3(350)
C = 2880 + 2205
C = 5085
Now, when Q is increased to 351, the annual inventory cost becomes:
C = 1,008,000/351 + 6.3(351)
C = 2870 + 2208.3
C ≈ 5078.3
The change in annual cost when Q is increased from 350 to 351 is approximately 5078.3 - 5085 = -6.7.
Now, let's find the instantaneous rate of change when Q = 350. We can do this by taking the derivative of the cost function with respect to Q:
dC/dQ = -1,008,000/Q^2 + 6.3
When Q = 350, the instantaneous rate of change is:
dC/dQ = -1,008,000/350^2 + 6.3
dC/dQ = -0.0816 + 6.3
dC/dQ ≈ 6.2184
So, the instantaneous rate of change when Q = 350 is approximately 6.2184.
Comparing this with the change in annual cost when Q is increased from 350 to 351, we see that -6.7 is not quite close to 6.2184. In fact, it's quite far apart! Maybe the manufacturer should stick to their original order size of 350, unless they want to take a tumble down the slippery slope of decreasing annual costs.
To find the change in annual cost when Q is increased from 350 to 351, we need to calculate the difference in annual cost for these two values of Q.
First, let's substitute Q = 350 into the cost equation to find the annual cost at Q = 350:
C = (1,008,00/Q) + 6.3Q
C = (1,008,00/350) + 6.3(350)
Now, let's calculate C when Q = 350:
C = 2,880 + 2,205
C = 5,085
So, the annual cost when Q = 350 is $5,085.
Next, let's substitute Q = 351 into the cost equation to find the annual cost at Q = 351:
C = (1,008,00/Q) + 6.3Q
C = (1,008,00/351) + 6.3(351)
Now, let's calculate C when Q = 351:
C = 2,872 + 2,206.3
C = 5,078.3
So, the annual cost when Q = 351 is $5,078.3.
To find the change in annual cost, we subtract the initial cost from the final cost:
Change in annual cost = Final cost - Initial cost
Change in annual cost = 5,078.3 - 5,085
Change in annual cost = -6.7
Therefore, the change in annual cost when Q is increased from 350 to 351 is -$6.7.
Now let's calculate the instantaneous rate of change when Q = 350. To do this, we need to find the derivative of the cost equation with respect to Q.
C = (1,008,00/Q) + 6.3Q
To find the derivative, we differentiate each term separately:
dC/dQ = (d/dQ)(1,008,00/Q) + (d/dQ)(6.3Q)
The derivative of 1,008,00/Q with respect to Q is -1,008,000/Q^2.
The derivative of 6.3Q with respect to Q is 6.3.
Therefore, the derivative of C with respect to Q is:
dC/dQ = -1,008,000/Q^2 + 6.3
To find the instantaneous rate of change when Q = 350, we substitute Q = 350 into the derivative:
dC/dQ = -1,008,000/350^2 + 6.3
Now we can calculate the value:
dC/dQ = -1,008,000/122500 + 6.3
dC/dQ ≈ -8.228
Therefore, the instantaneous rate of change when Q = 350 is approximately -8.228.
Comparing the change in annual cost (-$6.7) with the instantaneous rate of change (-8.228), we can see that the change in annual cost is a smaller rate of change compared to the instantaneous rate of change.
To find the change in annual cost when Q is increased from 350 to 351, you need to calculate the difference between the values of the annual cost function at these two order sizes.
First, substitute Q = 350 into the annual cost function C = 1,008,000/Q + 6.3Q:
C = 1,008,000/350 + 6.3(350) = 2880 + 2205 = 5085
Next, substitute Q = 351 into the annual cost function:
C = 1,008,000/351 + 6.3(351) = 2870 + 2208.3 = 5078.3
The change in annual cost when Q is increased from 350 to 351 is:
ΔC = 5078.3 - 5085 = -6.7
Therefore, the change in annual cost is -6.7.
To compare this with the instantaneous rate of change when Q = 350, we can calculate the derivative of the annual cost function with respect to Q at Q = 350.
The annual cost function is C = 1,008,000/Q + 6.3Q. To find the derivative, differentiate each term with respect to Q:
dC/dQ = -1,008,000/Q^2 + 6.3
Substitute Q = 350 into the derivative:
dC/dQ = -1,008,000/350^2 + 6.3 = -8.237
Therefore, the instantaneous rate of change when Q = 350 is -8.237.
Comparing the change in annual cost (-6.7) with the instantaneous rate of change when Q = 350 (-8.237), we can see that the change in annual cost is less than the instantaneous rate of change.