Final Answer:
The retailer should charge $28 for each candle to maximize its profit. Option C is answer.
Step-by-step explanation:
Let's analyze the problem step-by-step:
Step 1: Define Variables
Price per candle: $x
Profit per candle: p(x)
Number of candles sold at price $x: n(x)
Step 2: Define Relationships
We know that for each $2 decrease in price, 10 additional candles are sold. This relationship can be represented by:
n(x) = 100 + 5(x - 32)
We can also express the profit per candle as the difference between the price and the cost per candle (which we assume to be constant):
p(x) = x - c
Total profit is the product of the profit per candle and the number of candles sold:
Total profit = n(x) * p(x)
Step 3: Find the Price for Maximum Profit
To maximize profit, we need to find the value of x that maximizes the total profit function. This can be done by:
Finding the expression for total profit:
Total profit = (100 + 5(x - 32)) * (x - c)
Expanding the product:
Total profit = 100(x - c) + 5(x - 32)(x - c)
Differentiating the total profit expression with respect to x and setting it equal to zero to find the critical point:
d(Total profit)/dx = 5(x - 28) = 0
Solving for the critical point:
x = 28
Checking the second derivative to confirm that it's a maximum:
d^2(Total profit)/dx^2 = 10 > 0
Therefore, the price that maximizes profit is $28.
The retailer should charge $28 for each candle to maximize its profit.
Option C is answer.