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.