Math CentralQuandaries & Queries


Question from apoorva, a student:

During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that he lost two sales per day.

a. Find the demand function, assuming it is linear.
b. If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit?


The demand function is the price p(x) per necklace Terry can charge and if he sells x necklaces. If this is a linear function then

p(x) = ax + b where a and b are constants.

Last summer when the price was $10 he sold 20 necklaces per day so

$10 = p(20) = 20 a + b

When he increased the price to $10 + $1 = $11 he lost 2 sales per day. Thus

$11 = p(18) = 18 a + b

Solve these equations for a and b.

The revenue that Terry makes is R(x)= x × p(x) and his cost is C(x) = 6 x Write an expression for the profit he makes and use your knowledge of calculus to maximize the profit.


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