Page 231 - How To Solve Word Problems In Calculus

Solution
Let x = number of sinks ordered in each shipment.

C(x) = H(x) + O(x)

x 180
= 6 + 60
2 x
= 3x + 10,800x −1 x > 0

C (x) = 3−10,800x −2

10,800
0 = 3 −
x 2
10,800
= 3
x 2

2
3x = 10,800
2
x = 3600
x = 60
21,600
Since C (x) = 21,600x −3 = , it is clear that C (60) > 0.

x 3
Hence x = 60 corresponds to a relative minimum and, since it
is the only relative extremum for positive x, it represents the
absolute minimum. (This is typical for inventory problems
and this analysis may be omitted, if desired.)
Conclusion: The company should order sinks in lots of
60 to minimize total cost.

Note: You may have noticed that the actual cost of the sinks
was not considered in the previous example. The cost
is the same no matter what each lot size is and has
no bearing on the solution. Mathematically this cost,
obtained by multiplying the cost of each sink by 180,
is constant and will be zero when differentiated. This is
typical for inventory control problems.

EXAMPLE 13
A tire dealer buys 4000 tires a year from a local distributor.
Each tire costs $75, the ordering fee is $30 per shipment, and

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