Question

A student makes and sells necklaces at the beach during the summer months. The material for each necklace costs her $6 and she has been selling about 30 per day at $10
each. She has been wondering whether or not to raise the price, so she takes a survey and finds that for every dollar increase she would lose 4 sales a day,
(a) If she increases her pr by x dollars, then her price will be 10+x
produce the necklaces for a day will be 6(30-4x)
✓ dollars.
dollars, her number of sales for the day will be 30-(4x)
necklaces, and her cost to
(b) Write her profit P as a function of the number x of dollar increases in price:

Answer 1

The derivative of a function indicates how sensitive the function is to changes as the input variable value changes

The revenue, profit, maximum profit and price are;

(a) Her revenue on the day is 300 - 10·x - 4·x²

(b) Her profit function, P(x) = 120 + 14·x - 4·x²

(c) The price she should set the necklace is $11.75

(d) The maximum profit made is $132.25

The reasons the above values are correct are as follows:

The given parameters are:

Cost of material for each necklace =$6

The number of necklace she sells a day = 30

Selling price for each necklace when she sells 30 necklaces = $10

Number of sales lost for every dollar increase = 4 sales a day

(a) When the price is increased by x dollars, then we have;

New price = 10 + x

New cost of daily production, C = 6·(30 - 4·x) dollars

The number of sales for the day = 30 - 4·x

Her revenue for the day, R = (10 + x)·(30 - 4·x) = 300 - 10·x - 4·x²

(b) Her profit, P = Her daily revenue, R - Her daily production cost, C

Her cost function, P(x) = R - C

∴ P(x) = 300 - 10·x - 4·x² - 6·(30 - 4·x) = 120 + 14·x - 4·x²

Her profit as a function of the number of x dollar increase in price, is therefore, P(x) = 120 + 14·x - 4·x²

(c) To maximize profit, the derivative of the profit function is found and equated to zero as follows;

$\dfrac{dP(x)}{dx} = \dfrac{d}{dx} \left(120 + 14\cdot x - 4 \cdot x^2\right) = 14 - 8\cdot x$

x = 14/8 = 1.75

Therefore, the price she should set the necklace = 10 + 1.75 = 11.75

The price she should set the necklace = $11.75

(d) The maximum profit $P(x)_{max}$ = P(1.75) = 120 + 14×1.75 - 4×1.75² = 132.25

Her maximum profit, $P(x)_{max}$ = $132.25

Learn more about calculus and derivative here:

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