A band can sell 80 CDs at concert if they sell the Cds for $12 each. For every $1 increase in the price, they will sell 5 fewer Cds. Determine the price for a CD that will maximize the band's revenue.
Let x be the price.
The number sold is
N = 80 - 5(x -12)= 140 -5x.
The revenue is 140x -5x^2 dollars.
Try completing the square, since you probably don't know how to use differential calculus yet. Maximum revenue ($980) is at a price where
140 - 10x = 0
x = $14
At a price of $13 or $15, the revenue is $975.
is $14 by completing the square
To determine the price for a CD that will maximize the band's revenue, we need to find the price that will result in the highest number of CDs sold, since revenue is calculated by multiplying the price per CD by the number of CDs sold.
Let's break down the given information:
- The band can sell 80 CDs at a price of $12 each.
- For every $1 increase in price, the band will sell 5 fewer CDs.
We can represent the relationship between the price and the number of CDs sold using the equation:
Number of CDs sold = 80 - 5(price - 12)
To find the price that will maximize revenue, we need to find the maximum value of the product of the number of CDs sold and the price per CD.
Revenue = (Number of CDs sold) * (Price per CD)
Revenue = (80 - 5(price - 12)) * price
Now we can find the maximum value of revenue by finding the critical points. We take the derivative of the revenue equation and set it equal to 0:
d(Revenue)/d(price) = 0
80 - 5(price - 12) - 5 = 0
80 - 5(price - 12) = 5
-5(price - 12) = -75
price - 12 = 15
price = 27
Therefore, the price for a CD that will maximize the band's revenue is $27.