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?

Question

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?

Answer 1

If there are x subscribers over 3000, the profit function is

p(x) = (20 + .01x)(3000+x) for x>0
x = 900

Answer 2

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.