The profit in dollars in producing x items of some commodity is given by the equation P=−11x2+346.5x−2612.5.
How many items should be produced to break even?(If there are two break-even points, then enter the smaller value of x. Your solution may not be an integer. Use your calculator for irrational square roots.)
I know that P(x)=revenue - cost, but I don't know how to set revenue and cost equal to each other to find break-even if they are already combined in −11x2+346.5x−2612.5...
The function is P=−11x^2+346.5x−2612.5
break-even is where profit=0.
I think your equation already contains revenue and cost, since it is labeled as "profit"
I interpret "break-even" point as having no loss or profit, or , P = 0
so we want the roots of
-11x^2 + 346.5x - 2612.5
x = (-346.5 ± √(5112.25)/-22
= 12.5 or 19
Wolfram confirms my answers
http://www.wolframalpha.com/input/?i=−11x%5E2%2B346.5x−2612.5%3D0+
Oh yeah I forgot it meant that P = 0 thanks
To find the break-even point, you need to set the profit (P) equal to zero because at the break-even point, the revenue generated is equal to the cost.
In the given equation, the profit is represented by P, so the equation would be:
0 = -11x^2 + 346.5x - 2612.5
To solve this quadratic equation, you can use either factoring, completing the square, or the quadratic formula. Since factoring may not be straightforward in this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the equation, a = -11, b = 346.5, and c = -2612.5. Substituting these values into the quadratic formula:
x = (-(346.5) ± √((346.5)^2 - 4(-11)(-2612.5))) / (2(-11))
Now, let's simplify the equation further:
x = (-346.5 ± √(120102.25 - 114595)) / (-22)
x = (-346.5 ± √(5567.25)) / (-22)
x = (-346.5 ± 74.64) / (-22)
Now, you have two possible solutions:
x1 = (-346.5 + 74.64) / (-22)
x2 = (-346.5 - 74.64) / (-22)
Calculating these values:
x1 = 18.384
x2 = 195.818
Since the question asks for the smaller value, the number of items that should be produced to break even is approximately 18.384.
Remember to double-check your work and use a calculator for accurate calculations!