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)

Answer 1

The demand function is ( p(x) = 120 - x ). This linear model signifies that for every unit increase in price, Terry's daily necklace sales decrease by two units. The intercept, 120, represents the initial optimal price for maximizing sales.

Step-by-step explanation:

The demand function,
`\( p(x) = 120 - x \)`, is derived from Terry's necklace sales on the beach during the summer. Initially priced at $10 each with an average daily sale of 20 units, a subsequent price increase by $1 led to a reduction in the average sales by two units per day. This information suggests a linear relationship between price and quantity sold.

In a linear demand function, the slope represents the rate of change, and in this case, the slope is determined by the decrease in sales (2 units) divided by the increase in price ($1), resulting in a slope of -2. The intercept, 120, represents the initial price that Terry could charge to maximize sales.

The demand function ( p(x) = 120 - x ) encapsulates this relationship, where ( p(x) ) signifies the price and ( x ) denotes the quantity sold. The negative slope (-2) indicates that for every unit increase in price, the quantity sold decreases by two units. This linear model provides a clear and concise representation of how changes in price influence the quantity demanded, offering insights for Terry to optimize her pricing strategy during the summer months on the beach.

Answer 2

The demand function (price p as a function of units sold x) is
`\[ p(x) = -x + 12 \]`.

Step-by-step explanation:

Let
`\( p(x) \)` be the price function (in dollars) as a function of the number of necklaces sold
`\( x \)`. Since the demand function is assumed to be linear, it can be written in the form
`\( p(x) = mx + b \)` , where
`\( m \)` is the slope and
`\( b \)` is the y-intercept.

The problem states that when the price was $10, the sales averaged 20 per day. This point gives us the coordinates (20, 10) on the demand function.

Similarly, when the price was increased by $1, the average sales decreased by two per day. This corresponds to the point (18, 11) on the demand function.

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

`\[ m = \frac{{11 - 10}}{{18 - 20}} = -(1)/(2) \]`

Now that we have the slope, we can use one of the points (let's use
`(20, 10))` to find the y-intercept
`(\( b \)):`

`\[ 10 = -(1)/(2)(20) + b \]`

`\[ 10 = -10 + b \]`

`\[ b = 20 \]`

Therefore, the demand function is:

`\[ p(x) = -(1)/(2)x + 20 \]`

To make it more convenient, we can multiply through by 2 to eliminate the fraction:

`\[ p(x) = -x + 40 \]`

However, the demand function is typically written with a negative slope, so we can rewrite it as:

`\[ p(x) = -x + 12 \]`

This represents the demand function where
`\( p(x) \)` is the price in dollars as a function of the number of necklaces sold
`\( x \).`