The annual inventory cost C for a manufacturer is C=(1,008,000/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.
C(350) = 5085
C(351) = 5083
C'(Q) = -1008000/Q^2 + 6.3
C'(350) = -1.93
pretty close
To find the change in annual cost when Q is increased from 350 to 351, we can substitute these values into the given equation and calculate C for each value of Q.
For Q = 350:
C = (1,008,000/350) + 6.3(350)
C = 2,880 + 2,205
C = 5,085
For Q = 351:
C = (1,008,000/351) + 6.3(351)
C = 2,872 + 2,212.3
C = 5,084.3
To find the change in annual cost, we subtract the cost at Q=350 from the cost at Q=351:
Change in cost = C(Q=351) - C(Q=350)
Change in cost = 5,084.3 - 5,085
Change in cost = -0.7
The change in annual cost when Q increases from 350 to 351 is -0.7 units.
Now, let's find the instantaneous rate of change when Q=350. We can calculate the derivative of the given function to determine this.
C(Q) = (1,008,000/Q) + 6.3Q
Taking the derivative with respect to Q using the quotient and product rule, we have:
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
dC/dQ ≈ -2.467
The instantaneous rate of change when Q=350 is approximately -2.467 units.
Comparing the change in annual cost when Q is increased from 350 to 351 (-0.7 units) with the instantaneous rate of change when Q=350 (-2.467 units), we can see that the change in annual cost is less than the instantaneous rate of change.
To find the change in annual cost when Q is increased from 350 to 351, we need to calculate the difference between the annual costs at these two values of Q.
Step 1: Calculate the annual cost when Q is 350.
Substitute Q = 350 into the given equation: C = (1,008,000/350) + 6.3(350)
Evaluate this expression to find the annual cost when Q = 350.
Step 2: Calculate the annual cost when Q is 351.
Substitute Q = 351 into the equation: C = (1,008,000/351) + 6.3(351)
Evaluate this expression to find the annual cost when Q = 351.
Step 3: Find the change in annual cost.
Calculate the difference between the annual costs at Q = 351 and Q = 350. This will give you the change in annual cost when Q is increased by 1.
To compare this with the instantaneous rate of change when Q = 350, we need to find the derivative of the cost function with respect to Q and evaluate it at Q = 350. The derivative will give us the instantaneous rate of change of the cost function with respect to Q.
To find the derivative of the cost function, differentiate each term separately and simplify the equation. Then substitute Q = 350 into the derivative expression. Evaluate this expression to find the instantaneous rate of change when Q = 350.
Finally, compare the change in annual cost (from Step 3) with the instantaneous rate of change (from the derivative evaluation) when Q = 350. This will allow you to determine the relationship between the two quantities.