Page 229 - How To Solve Word Problems In Calculus

Step 2
The total profit to the store is
P(x) = (profit per shirt) × (number of shirts sold).

Step 3
Since the cost of the shirt is $25, the store’s profit per
shirt is (40 − x) − 25 = 15 − x. The number of shirts sold at
this price is 55 + 5x since 5 more shirts are sold for each dollar
decrease in price.

P(x) = (15 − x)(55 + 5x)
= 825 + 20x − 5x 2

Step 4

P (x) = 20 − 10x
0 = 20 − 10x

10x = 20
x = 2

Step 5
(optional) P (x) =−10 for all x. Since the second deriva-

tive is negative, the critical number represents a relative maxi-
mum. Since there is only one relative extremum, the absolute
maximum profit occurs when x = 2.
The optimal selling price is $40 − $2 = $38.

Inventory Control
When a company orders spare parts or raw materials, two spe-
cific costs are incurred.

(a) The company must pay an ordering fee to cover handling
and transportation costs.
(b) The company must pay a storage fee to cover the cost of
keeping the items on hand until needed. These costs are
known as storage or holding costs.
If each shipment is large, fewer shipments will be needed and
consequently the ordering fees will be small. However, hold-
ing costs will be higher. If many small shipments are made,

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