A manufacture finds the total cost “C” for a product is expressed by C=3x2-8, and the total revenue
“R” by R=3x-2, where x is the number of units sold. What is the breakeven point (where total
cost=total revenue) to the nearest unit?
3x^2 - 8 = 3x - 2
3x^2 - 3x - 6 = 0
x^2 - x - 2 = 0
(x-2)(x+1) = 0
x = 2 or x = -1, but you can't make negative units, so
x = 2
To find the breakeven point, we need to set the total cost equal to the total revenue and solve for the value of x.
Given:
Total Cost (C) = 3x^2 - 8
Total Revenue (R) = 3x - 2
Equating the two:
3x^2 - 8 = 3x - 2
Rearranging the equation:
3x^2 - 3x - 6 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = -3, and c = -6.
Plugging in the values into the quadratic formula:
x = (-(-3) ± √((-3)^2 - 4 * 3 * -6)) / (2 * 3)
x = (3 ± √(9 + 72)) / 6
x = (3 ± √81) / 6
x = (3 ± 9) / 6
Simplifying further:
x = (3 + 9) / 6 = 12 / 6 = 2
x = (3 - 9) / 6 = -6 / 6 = -1
The breakeven point is the value of x where the total cost equals the total revenue. In this case, x = 2 is the breakeven point to the nearest unit.