During the summer months Terry makes and sells necklaces on...

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 the average decreased by two sales per day. $\begin{array}{c}{\text { (a) Find the demand function, assuming that it is linear. }} \\ {\text { (b) If the material for each necklace costs Terry } \$ 6, \text { what }} \\ {\text { should the selling price be to maximize his profit? }}\end{array}$

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 the average decreased by two
sales per day.
$\begin{array}{c}{\text { (a) Find the demand function, assuming that it is linear. }} \\ {\text { (b) If the material for each necklace costs Terry } \$ 6, \text { what }} \\ {\text { should the selling price be to maximize his profit? }}\end{array}$

Key Concepts

Demand Function

The demand function models the relationship between the price of a product and the quantity of the product that consumers are willing and able to purchase. In many cases, particularly when assuming a linear form, it expresses the quantity demanded as a linear function of price, enabling predictions about how changes in price can affect sales volume.

Linear Modeling

Linear modeling involves representing relationships between variables using a linear equation. In the context of demand, it simplifies the analysis by suggesting that there is a constant rate of change in quantity with respect to price, which is often derived from observed changes under different pricing conditions.

Profit Function

The profit function calculates the net profit by subtracting total costs from total revenue. It is constructed by considering the revenue generated from sales (price multiplied by quantity demanded) and subtracting costs associated with producing or acquiring the products. This function is crucial for determining the pricing strategy that maximizes profit.

Profit Maximization

Profit maximization is the process of finding the price or production level that results in the highest possible profit. When the profit function is quadratic, this can be achieved by analyzing the vertex of the parabola, which represents the optimal point where revenue minus cost is maximized. This approach is fundamental in economic decision-making and pricing strategies.

Transcript

00:01 For this problem, we are looking at a necklace producer named terry who sells his necklaces at the beach.

00:08 So we're going to be looking at the demand or the price function.

00:13 And we're going to look at selling price to see if we can figure out how to maximize his profit.

00:18 First, let's see what we know.

00:20 We know that he makes a certain number of necklaces.

00:24 So we're going to let x equal the number of necklaces that terry produces.

00:29 And p of x is what we typically use for the demand or the price function.

00:34 So p of x is going to be the price per necklace.

00:40 Okay.

00:41 So what do we know? we know if he sells a necklace for $10, he can get about $20 per day that he sells.

00:48 So 20 necklaces per day when the price is $10 a piece.

00:54 If i increase the price by a dollar, so now my price is $11, i'm told that he decreases the number that he sells by two.

01:03 So increase to $11, i'm only selling $18.

01:07 So that is the information we're given.

01:11 Now, if we assume that this correlation between the numbers sold and the price, if we assume that this is linear, we can find the slope of this line.

01:20 So let's do that.

01:21 The difference in my ys, that'll be 10 minus 11, over the difference in my x's, or 20 minus 18.

01:29 That's going to be negative 1 half.

01:32 There's my slope.

01:34 Now, if i know a point and a slope, which i do, i should know two points in a slope, i can use the point slope form, which says y minus y1 equals m times x minus x1.

01:47 Or another way to write this, i can move that y1 to the left hand side or to the right hand side.

01:53 So i get y equals m times x minus x1 plus y1.

01:58 So we're going to use this form for what we have here.

02:03 Y, which is my output, in this case, my output's p of x.

02:06 That's the price per necklace.

02:08 That's what i'm trying to find.

02:09 Y equals my slope, which is negative one half, times x minus.

02:15 Now i need a point.

02:16 I'm going to take the first point just because because it's first, you'll get the exact same answer no matter which point you pick.

02:23 I'm going to say x minus x1, which is 20, plus y1, which is 10.

02:30 And i'm going to get rid of these parentheses.

02:32 So that becomes negative 1 .5 .x...