The county news earns a profit of $20 per year for each of its 3000 subscribers. Management projects that the profit per subscriber would increase by $0.01 for each additional subscriber over the current 3,000. How many subscribers are needed to bring a total profit of $113,100?
If there are x subscribers over 3000, the profit function is
p(x) = (20 + .01x)(3000+x) for x>0
x = 900
To find the number of subscribers needed to bring a total profit of $113,100, we can set up an equation.
Let's denote the number of additional subscribers as "x". According to the given information, the profit per subscriber increases by $0.01 for each additional subscriber. Therefore, the profit per subscriber is $20 + ($0.01 * x).
The total profit from the additional subscribers can be calculated as ($0.01 * x * number of subscribers):
= $0.01 * x * (3000 + x)
And the equation for total profit is:
Total profit = Profit from existing subscribers + Profit from additional subscribers
$113,100 = ($20 * 3000) + ($0.01 * x * (3000 + x))
Simplifying the equation:
$113,100 = $60,000 + $0.01x^2 + $30x
Rearranging the equation to form a quadratic equation:
$0.01x^2 + $30x + ($60,000 - $113,100) = 0
$0.01x^2 + $30x - $53,100 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 0.01, b = 30, and c = -53,100.
Using the quadratic formula:
x = (-30 ± √(30^2 - 4 * 0.01 * -53,100)) / (2 * 0.01)
Simplifying further:
x = (-30 ± √(900 + 2,124)) / 0.02
x = (-30 ± √3,024) / 0.02
Now, we can calculate the two possible values for "x":
x1 = (-30 + √3,024) / 0.02
x2 = (-30 - √3,024) / 0.02
Note that the negative value for "x" is not meaningful in this context since we cannot have a negative number of additional subscribers. So let's only consider the positive value for "x".
x = (-30 + √3,024) / 0.02
After calculating this value, we will have the number of additional subscribers needed to bring a total profit of $113,100. Adding this value to the existing number of subscribers (3,000), we will have the final answer.