A news papers profit is $25 per year for each of it's 5500 subscribers. Management estimates that the profit per subscriber will increase by 1 cent for each additional subscriber over the current 5500.How many will bring a total profit of $180,000
To solve this problem, we can use algebraic equations.
Let's start by defining the variables:
- P: Profit per subscriber
- N: Number of subscribers
- T: Total profit
According to the given information, we know that the profit per subscriber is $25 for the initial 5500 subscribers. Additionally, we know that for each additional subscriber, the profit per subscriber will increase by 1 cent.
So, we can create the following equation to represent the profit per subscriber:
P = 25 + 0.01 * (N - 5500)
Now, we need to express the total profit in terms of the number of subscribers. In this case, the total profit is given as $180,000. So, we can set up the following equation:
T = P * N
Substituting the expression for P from the first equation into the second equation, we get:
T = (25 + 0.01 * (N - 5500)) * N
Now, we can solve this equation to find the number of subscribers that will bring a total profit of $180,000.
180,000 = (25 + 0.01 * (N - 5500)) * N
Simplifying the equation further, we can expand the expression:
180,000 = 25N + 0.01N^2 - 277.5N
Combining like terms, we have a quadratic equation:
0.01N^2 - 252.5N + 180,000 = 0
To solve this quadratic equation, we can use the quadratic formula:
N = (-b ± √(b^2 - 4ac)) / 2a
In our case, the quadratic equation is in the form of ax^2 + bx + c = 0, with the following values:
a = 0.01
b = -252.5
c = 180,000
Plugging in these values, the quadratic formula becomes:
N = (-(-252.5) ± √((-252.5)^2 - 4 * 0.01 * 180,000)) / (2 * 0.01)
Simplifying further, we have:
N = (252.5 ± √(64,002.25 - 7,200)) / 0.02
N = (252.5 ± √56,802.25) / 0.02
N = (252.5 ± 238.5) / 0.02
Now, we can solve for both possible values of N:
N1 = (252.5 + 238.5) / 0.02 = 24350
N2 = (252.5 - 238.5) / 0.02 = 7000
Therefore, to bring a total profit of $180,000, the newspaper needs either 24,350 subscribers or 7,000 subscribers.