Question

A company's profit when it sells X thousand items is predicted to be P(x) = -3x^2 + 1278x - 1000.

A) what is the company's startup costs? Answer: $10,000
B) what is the least number of items the company need to sell to break even? (to the nearest thousand items) _______ thousand items
C) how many items should the company sell to maximize profit? (to the nearest thousand items) Answer: 213 thousand items
Please help answer part B.

Answer 1

To find the least number of items the company needs to sell to break even, we need to find where the profit is zero. In other words, we need to solve the equation:

-3x^2 + 1278x - 1000 = 0

We can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = -3, b = 1278, and c = -1000.

Plugging in these values, we get:

x = (-1278 ± sqrt(1278^2 - 4(-3)(-1000))) / 2(-3)
x = (-1278 ± sqrt(1638884)) / (-6)

Simplifying, we get:

x = (-1278 ± 1280.22) / (-6)

x = (-1278 + 1280.22) / (-6) or x = (-1278 - 1280.22) / (-6)

x = 0.37 thousand items or x = 213.04 thousand items (rounded to two decimal places)

We can ignore the first solution as it doesn't make sense to sell 0.37 thousand items and hence, the company needs to sell roughly 213 thousand items to break even.