Question

During the summer months Terry makes and sells necklaces on the beach. Last summer she sold the necklaces for $10 each, and her sales averaged 20 per day. When she increased the price by $1, she found that the average decreased by two sales per day. (a) Find the demand function (price p as a function of units sold x), assuming that it is linear. p(x) = (b) If the material for each necklace costs $8, what selling price should Terry set to maximize her profit? per necklace

Answer 1

To find the demand function, we use the given information about the change in average sales when the price is increased. If the demand function is assumed to be linear, we can find the slope and y-intercept to get the demand equation. To maximize profit, Terry needs to calculate the profit for each selling price and determine the selling price that yields the highest profit. However, without more information about production costs and other factors, we can't determine the specific selling price that will maximize profit.

Step-by-step explanation:

To find the demand function, we can use the given information about how the average sales change when the price is increased. Let's assume the demand function is linear and can be represented as p(x) = mx + b, where p is the price and x is the number of necklaces sold.

From the information given, we know that when the price was $10, the sales averaged 20 per day. This means we have the point (20, 10) on the demand line. When the price was increased by $1, the average sales decreased by two per day. This gives us another point on the demand line: (18, 11).

Using these two points, we can find the slope (m) of the demand line:

m = (11 - 10) / (18 - 20) = 1 / -2 = -1/2

Now we can use the slope and one of the points to find the y-intercept (b):

10 = (-1/2)(20) + b

10 = -10 + b

b = 20

Therefore, the demand function is p(x) = -1/2x + 20.

To maximize profit, Terry needs to find the selling price that will result in the highest profit margin. To do this, she needs to calculate the profit for each possible selling price by subtracting the cost of materials ($8) from the selling price, and then determine which selling price yields the highest profit.

Let's assume the selling price is p. The profit for each necklace sold can be calculated as:

Profit = p - $8

Since Terry sells x necklaces per day, her total profit will be:

Total Profit = (p - $8)x

To find the selling price that maximizes profit, Terry needs to determine the value of p that maximizes the total profit function.

However, we don't have enough information to find the exact value of p without additional information about the cost of production, overhead costs, and other factors that determine profit. Without this additional information, we can't determine the specific selling price that will maximize Terry's profit.

Answer 2

The demand function given the linear assumption is p(x) = -0.5x + 20. To maximize profit, the cost function and revenue function must be considered, and by setting the derivative of the profit function to zero and solving, the optimal price point for maximum profit can be found.

Step-by-step explanation:

To find the demand function, we start by recognizing two points on the demand curve: (20, 10) and (18, 11), where 20 and 18 are the numbers of necklaces sold per day (x), and 10 and 11 are the corresponding prices (p). Since we assume the demand function is linear, we can use the two points to generate the equation of a line in the form p(x) = mx + b.

Using the two given points:

1. Calculate the slope (m) of the line: m = (p2 - p1) / (x2 - x1) = (11 - 10) / (18 - 20) = 1 / -2 = -0.5.
2. We can now insert one of the points into the equation p = mx + b and solve for b, the y-intercept: 10 = (-0.5)(20) + b, which gives us b = 20.
3. The demand function is therefore p(x) = -0.5x + 20.

To maximize profit, we must consider both the cost of material and the price point that maximizes revenue. The material cost per necklace is $8. We must find the selling price that maximizes profit, knowing that profit is revenue minus cost.

Let C(x) represent the cost function, which is C(x) = 8x, and R(x) represent the revenue function, which is R(x) = xp(x). To maximize profit, we need to maximize the profit function P(x) = R(x) - C(x) = x(-0.5x + 20) - 8x. Taking the derivative of P(x) with respect to x and setting it to zero will give us the quantity x that maximizes profit. Using the quadratic formula or completing the square will result in the optimal price point.