Terry makes and sells necklaces. He has observed over time that when the price is $12 each, he sells an average of 20 per day. If he increases the price, then his average sales fall by 2 per day for each dollar increase. The materials for each necklace cost $7. Express his profit P as a function of x, the number of necklaces sold.

Question

Terry makes and sells necklaces. He has observed over time that when the price is $12 each, he sells an average of 20 per day. If he increases the price, then his average sales fall by 2 per day for each dollar increase. The materials for each necklace cost $7. Express his profit P as a function of x, the number of necklaces sold.

that would make p(x)= ?

THANK YOU!

Answer 1

at price z, demand

x = 20-2(z-12)
or, price z = 22 - x/2

p(x) = revenue-cost
= x(22 - x/2) - 7x
= 15x - 1/2 x^2

Answer 2

To express Terry's profit P as a function of x, the number of necklaces sold, we can calculate the revenue and subtract the cost.

Let's break it down step by step:

1. Determine Terry's revenue:
- We know that when the price is $12 each, he sells an average of 20 per day.
- So, his revenue for selling 20 necklaces at $12 each is 20 * 12 = $240.

2. Calculate the number of necklaces sold, x:
- Since Terry's average sales fall by 2 per day for each dollar increase in price, we can find x by subtracting the number of sales lost due to price increase from the initial sales of 20.
- If the price is increased by $1, the average sales will decrease by 2 per day.
- Therefore, if the price is increased by $1, Terry will sell 20 - 2 = 18 necklaces per day.
- So, for each dollar increase in price, the number of necklaces sold per day decreases by 2.
- Mathematically, we can express x (number of necklaces sold) as: x = 20 - 2(p - 12), where p is the price.

3. Calculate Terry's cost:
- Since the materials for each necklace cost $7, Terry's total cost for x necklaces sold will be 7x.

4. Calculate Terry's profit:
- To calculate the profit, we subtract the cost from the revenue: P = Revenue - Cost.
- P(x) = (Price * Number of necklaces sold) - Cost
- P(x) = (p * x) - 7x.

So, the expression for Terry's profit, P(x), is:
P(x) = (p * x) - 7x

Keep in mind that this function assumes a linear relationship between the price, the number of necklaces sold, and the profit. In reality, various factors can influence sales and profit, so this function might not represent the entire picture accurately.