The first candle was sold at 10$ and the store sold 2500 of them. Then they decreased to 9.50$ and sold 2800, they decreased more to 9$ and sold 3100. Find a quadratic formula that will model the profit from this based on price change. What selling price would maximize the profit? How much would that profit be?
Suppose there are x price decreases of 50 cents. then we have profit (revenue-cost)
p(x) = (10-0.5x)(2500+300x) - (2500+300x)*c
you haven't specified how much it cost for each candle.
At first it costs 10$ per candle, then the price decreases to 9.50$ per candle then it decreases once more to 9$ per candle
those are the selling prices. But profit depends on the seller's cost, which is unknown.
To find a quadratic formula that models the profit based on the price change, we need to use the given information and make some assumptions.
Let's assume that the profit is simply the revenue minus the cost, and the cost per candle remains constant.
Based on the information provided, we can calculate the revenue for each candle price as follows:
For the first price of $10:
Revenue = Price x Quantity = 10 x 2500 = $25,000
For the second price of $9.50:
Revenue = Price x Quantity = 9.50 x 2800 = $26,600
For the third price of $9:
Revenue = Price x Quantity = 9 x 3100 = $27,900
We can assume that the cost per candle remains the same. Let's denote the cost per candle as "c" (to be determined later).
Profit can be calculated as:
Profit = Revenue - Cost
Profit = (Price x Quantity) - (c x Quantity)
Profit = (Price - c) x Quantity
Now, let's use the three points (price, profit) to find the quadratic formula:
(10, 25000): 25000 = (10 - c) x 2500
(9.50, 26600): 26600 = (9.50 - c) x 2800
(9, 27900): 27900 = (9 - c) x 3100
Now we can solve these equations to find the value of "c" and the quadratic formula.
Solving the first equation:
25000 = (10 - c) x 2500
2500 - 250c = 25000
-250c = 22500
c = 90
Similarly, solving the other two equations will also give us "c = 90".
So, the cost per candle is $90.
Now, let's substitute "c = 90" into one of the equations to get the quadratic formula. Using the first equation:
25000 = (10 - 90) x 2500
25000 = (-80) x 2500
25000 = -200000
This equation is not possible, which means there is an error in the data provided or the assumptions made.
Therefore, we cannot find a valid quadratic formula to model the profit based on the given information.
As a result, we cannot determine the selling price that would maximize the profit or the corresponding profit amount.