An electronics store is selling personal CD players. The regular price for each
CD player is $90. During a typical two weeks, the store sells 50 units. Past
sales indicate that for every $1 decrease in price, the store sells five more
units during two weeks. Calculate the price that will maximize revenue
revenue = price * units
x = the price decrease
r = (90 - x) (50 + 5x) = 4500 + 400 x - 5 x^2
the max is on the axis of symmetry ... where the 1st derivative equals zero
dr/dx = 400 - 10 x
400 - 10 x = 0 ... x = 40
price for max revenue = 90 - 40
Well, it sounds like you're in a bit of a pricing pickle. Let's see if we can find a solution that's music to your ears.
We know that for every $1 decrease in price, the store sells five more units. So, if we decrease the price by $1, we can expect to sell 55 units (50 + 5) during the two weeks.
To maximize revenue, we need to find the sweet spot where the number of units sold multiplied by the price is highest. Let's break it down:
If we decrease the price by $1, the new price would be $89, and we can expect to sell 55 units, resulting in a revenue of $4,895 (89 * 55).
If we decrease the price by $2, the new price would be $88, and we can expect to sell 60 units, resulting in a revenue of $5,280 (88 * 60).
Continuing this pattern, if we decrease the price by $3, the new price would be $87, and we can expect to sell 65 units, resulting in a revenue of $5,655 (87 * 65).
Let's keep going and crunch the numbers:
If we decrease the price by $4, the new price would be $86, and we can expect to sell 70 units, resulting in a revenue of $6,020 (86 * 70).
If we decrease the price by $5, the new price would be $85, and we can expect to sell 75 units, resulting in a revenue of $6,375 (85 * 75).
Based on these calculations, it seems that decreasing the price by $5 to a selling price of $85 will maximize your revenue at $6,375.
So, there you have it – a price that will make your customers groove with joy while keeping your revenue rolling in. Happy selling!
To calculate the price that will maximize revenue, we need to determine the price point at which the number of units sold is maximized. Revenue is calculated by multiplying the price by the number of units sold.
Let's start by finding the relationship between the price and the number of units sold. We know that for every $1 decrease in price, the store sells five more units.
Let's assume that the price is decreased by $x, resulting in a new price of $90 - $x. The number of units sold will increase by 5x.
Therefore, the number of units sold can be represented by the equation:
units sold = 50 + 5x
Now, let's calculate the revenue. Revenue is given by the equation:
revenue = price * units sold
Substituting the price and units sold equations, we get:
revenue = (90 - x) * (50 + 5x)
To find the price that maximizes revenue, we need to find the value of x that maximizes the revenue equation.
Let's expand and simplify the revenue equation:
revenue = (90 - x) * (50 + 5x)
revenue = 4500 - 50x + 450x - 5x^2
revenue = -5x^2 + 400x + 4500
To find the value of x that maximizes the revenue, we can use calculus. Taking the derivative of the revenue equation and setting it equal to zero will give us the critical point.
Let's differentiate the revenue equation:
d(revenue)/dx = -10x + 400
Setting the derivative equal to zero and solving for x:
-10x + 400 = 0
-10x = -400
x = 40
Therefore, when x = 40, the price decrease that maximizes revenue is $40.
Substituting x = 40 back into the revenue equation:
revenue = -5(40)^2 + 400(40) + 4500
revenue = -8000 + 16000 + 4500
revenue = $22,500
So, the price that will maximize revenue is $90 - $40 = $50, and the maximum revenue will be $22,500.
To calculate the price that will maximize revenue, we need to find the price that maximizes the product of the price and the number of units sold. This can be done by using the concept of elasticity of demand.
We know that for every $1 decrease in price, the store sells five more units. This means that the price elasticity of demand is -5, as a decrease in price leads to an increase in quantity demanded.
To find the price that will maximize revenue, we need to find the price at which the elasticity of demand is equal to -1. At this price, a decrease in price will result in a proportional increase in quantity demanded, leading to maximum revenue.
Let's solve for the price:
Original price: $90
Let the decrease in price be x.
New price: $90 - x
Number of units sold when price decreases by 1: 50 + 5(1) = 55
According to the price elasticity formula:
Elasticity of demand (E) = percentage change in quantity demanded / percentage change in price
Using the midpoint formula, we can calculate the price elasticity as:
E = (change in quantity demanded / average quantity demanded) / (change in price / average price)
E = (5 / 52.5) / (1 / 90)
E = (5 * 90) / (1 * 52.5)
E = 450 / 52.5
E ≈ -8.57
Since the elasticity of demand is -8.57, we need to find the price at which the elasticity becomes -1 (more elastic).
Let the decrease in price be y (the price at which the elasticity becomes -1).
New price: $90 - y
Number of units sold when price decreases by y: 50 + 5y
Again, using the price elasticity formula:
E = (change in quantity demanded / average quantity demanded) / (change in price / average price)
Setting E to -1, we can solve for y:
-1 = (5y / (50 + 5y)) * (90 / y)
Simplifying the equation:
-1 = (5(90)) / (50 + 5y)
-1 = 450 / (50 + 5y)
Cross-multiplying and re-arranging the equation:
-50 - 5y = 450
5y = -500
y = -100
So, when the price decreases by $100, the elasticity of demand becomes -1.
To find the price that will maximize revenue, we subtract the value of y from the original price:
Price that maximizes revenue = $90 - $100 = -$10
However, a negative price is not meaningful in this context. Therefore, the price that will maximize revenue is $90.