The Vilas County News earns a profit of $20 per year for each of its 3,000 subscribers. Management projects that the profit per subscriber would increase by 1¢ for each additional subscriber over the current 3,000. How many subscribers are needed to bring a total profit of $153,525?
If you mean the increase occurs only for those in excess of 3000, then you have
(20)(3000) + (20+0.01)(x-3000) = 153525
7,674
To find the number of subscribers needed to bring a total profit of $153,525, we can set up an equation.
Let's denote the number of additional subscribers as x. The profit per subscriber would increase by 1¢ for each additional subscriber, so the profit per subscriber would be $20 + $0.01x.
The total profit can be calculated by multiplying the profit per subscriber by the number of subscribers. The equation is:
($20 + $0.01x) * (3,000 + x) = $153,525
To solve this equation, we can distribute and then simplify:
60,000 + 20x + 3x + 0.01x^2 = 153,525
0.01x^2 + 23x + 60,000 = 153,525
0.01x^2 + 23x + 60,000 - 153,525 = 0
0.01x^2 + 23x - 93,525 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 0.01, b = 23, and c = -93,525. Plugging in these values, we can calculate:
x = (-23 ± √(23^2 - 4 * 0.01 * -93,525)) / (2 * 0.01)
x = (-23 ± √(529 + 3,741)) / 0.02
x = (-23 ± √4,270) / 0.02
There are two possible solutions:
1. x = (-23 + √4,270) / 0.02
2. x = (-23 - √4,270) / 0.02
Calculating each solution:
1. x ≈ 1080
2. x ≈ -1098
Since the number of subscribers cannot be negative, we discard the second solution.
Therefore, the number of subscribers needed to bring a total profit of $153,525 is approximately 3,000 + 1,080 = 4,080 subscribers.
To find the number of subscribers needed to bring a total profit of $153,525, we need to break down the problem step by step.
Step 1: Calculate the profit for the current 3,000 subscribers.
The Vilas County News earns a profit of $20 per year for each of its 3,000 subscribers.
Profit for 3,000 subscribers = $20 x 3,000 = $60,000
Step 2: Determine the additional profit for each additional subscriber.
Management projects that the profit per subscriber would increase by 1¢ for each additional subscriber over the current 3,000.
So, the additional profit per subscriber = $0.01
Step 3: Calculate the total additional profit for all the additional subscribers.
To determine the total additional profit, we need to find the number of additional subscribers. Let's assume the number of additional subscribers needed is "x".
Since the profit increases by 1¢ for each additional subscriber, the total additional profit for "x" subscribers can be calculated as:
Total additional profit = $0.01 x x
Step 4: Calculate the total profit for the required number of subscribers.
The total profit is the sum of the profit for the current subscribers and the additional profit from the additional subscribers.
Total profit = Profit for 3,000 subscribers + Total additional profit
Total profit = $60,000 + $0.01x
Step 5: Write the equation and solve for "x".
Given that the total profit needed is $153,525, we can set up the equation:
$153,525 = $60,000 + $0.01x
Now, let's solve the equation:
$0.01x = $153,525 - $60,000
$0.01x = $93,525
x = $93,525 / $0.01
x = 9,352,500
Therefore, the number of subscribers needed to bring a total profit of $153,525 is 9,352,500.