Answer:
Step-by-step explanation:
We know that Julian's profit per download is 0.75x dollars. Let's start by setting up an equation for his total profit, given the number of downloads:
Total profit = (profit per download) x (number of downloads)
We also know that the number of downloads Julian will sell is given by the equation:
Number of downloads = -4x + 160
So we can substitute this expression for the number of downloads into the equation for total profit:
Total profit = (profit per download) x (-4x + 160)
Simplifying this expression:
Total profit = -3x(4x - 160)
Total profit = -12x^2 + 480x
To earn exactly $900 in profit, we set the total profit equal to 900 and solve for x:
-12x^2 + 480x = 900
Dividing both sides by -12:
x^2 - 40x + 75 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in a = 1, b = -40, and c = 75:
x = (-(-40) ± sqrt((-40)^2 - 4(1)(75))) / 2(1)
x = (40 ± sqrt(400)) / 2
x = 20 ± 10
So the two prices Julian can charge per download to earn exactly $900 in profit in his first year are $10 and $30.
To verify, we can calculate Julian's profit for each price:
At $10 per download, Julian will sell -4(10) + 160 = 120 downloads. His profit per download is 0.75(10) = $7.50, so his total profit will be 120 x 7.50 = $900.
At $30 per download, Julian will sell -4(30) + 160 = 40 downloads. His profit per download is 0.75(30) = $22.50, so his total profit will be 40 x 22.50 = $900.