Final answer:
To maximize daily revenue, the online music store should charge $1.75 per song. This is found by creating a revenue function from the given linear relationship between price and quantity sold and then finding the vertex of the resulting quadratic equation.
Step-by-step explanation:
To solve this problem, we need to find the price per song that maximizes the online music store's daily revenue.
Let's call the price per song p, and let the number of songs sold be x. Initially, the store sells 4000 songs a day at $1 each, so the revenue is $4000.
If the price increases by $0.05, the number of songs sold decreases by 80.
The price increase can be represented as p = 1 + 0.05n, where n is the number of $0.05 increases.
Each increase in price reduces the number of songs sold, which can be modeled as x = 4000 - 80n.
The revenue function R can be described by R = p × x, which becomes R = (1 + 0.05n)(4000 - 80n).
Expanding the revenue function and then converting it to a standard quadratic equation provides
R = 4000 + 200n - 80n - 4n2 or R
= -4n2 + 120n + 4000.
To maximize revenue, we can find the vertex of the parabola represented by this quadratic equation.
The profit-maximizing price is at the vertex of the parabola, which occurs at n = -b/2a where a is the coefficient of n2 and b is the coefficient of n.
Plugging in the values from the quadratic equation
we get n = -120/(2 × -4)
= 15.
Once we've found the optimal number of price increases, we can find the corresponding price:
p = 1 + 0.05 × 15
= $1.75.
Therefore, the store should charge $1.75 per song to maximize daily revenue.