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?
To find the breakeven point, we need to set the total cost equal to the total revenue and solve for x.
Given that the total cost, C, is expressed as C = 3x^2 - 8, and the total revenue, R, is expressed as R = 3x - 2, we can equate the two:
3x^2 - 8 = 3x - 2
Now, let's solve this equation to find the value of x.
First, we can rearrange the equation to bring all terms to one side:
3x^2 - 3x - 6 = 0
Next, we can simplify the equation by factoring if possible. In this case, factoring is not straightforward, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = -3, and c = -6.
Plugging these values into the quadratic formula, we get:
x = (-(-3) ± √((-3)^2 - 4(3)(-6))) / (2(3))
Simplifying further:
x = (3 ± √(9 + 72)) / (6)
x = (3 ± √81) / 6
x = (3 ± 9) / 6
x = (3 + 9) / 6 or x = (3 - 9) / 6
x = 12 / 6 or x = -6 / 6
x = 2 or x = -1
Since we are looking for the breakeven point in terms of units sold, we can discard the negative value (-1) as it is not meaningful in this context. Therefore, the breakeven point is x = 2 to the nearest unit.