Question

A candlemaker prices one set of scented candles at $10 and sells an average of 200 sets each week. He finds that when he reduces the price by $1, he then sells 50 more candle sets each week. A function can be used to model the relationship between the candlemaker's weekly revenue, R(x), after x one-dollar decreases in price.

Answer 1

R(x) = -50x2 + 300x + 2000

Maximum revenue of $2450 with x = 3

Explanation:

For each 1$ reduced of the price, he sells 50 more candle sets, so if he reduces x times the price by 1$, the final price will be:

price = 10 - x

And the number of sets sold will be:

sets = 200 + 50x

If the revenue is the price times the number of sets sold, we have that:

R(x) = (10 - x) * (200 + 50x)

R(x) = 2000 + 500x - 200x - 50x2

R(x) = -50x2 + 300x + 2000

To find the value of x that gives the maximum revenue, we use the formula of the vertex of the quadratic equation:

x_v = -b / 2a

x_v = -300 / (-100) = 3

And the maximum revenue is:

R(x_v) = -50*3^2 + 300*3 + 2000 = $2450