If a company has total costs C(x) = 20,000 + 25x + 0.2x2 and total revenues given by R(x) = 475x − 0.8x2, find the break-even points.
To find the break-even points, we need to determine the value(s) of x where the total costs and total revenues are equal.
First, let's set up the equation for the break-even point:
C(x) = R(x)
Substituting the given equations:
20,000 + 25x + 0.2x^2 = 475x - 0.8x^2
Combining like terms:
0.2x^2 + 0.8x^2 + 25x - 475x - 20,000 = 0
Simplifying:
x^2 + 400x - 20,000 = 0
Now, we can solve this quadratic equation for the values of x using either factoring, completing the square, or the quadratic formula.
Since factoring may not be the most efficient method in this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-(400) ± √((400)^2 - 4(1)(-20,000))) / (2(1))
Calculating:
x = (-400 ± √(160000 + 80,000)) / 2
x = (-400 ± √(240,000)) / 2
x = (-400 ± 489.8979) / 2
Now, let's evaluate both possibilities:
x1 = (-400 + 489.8979) / 2
x1 = 89.8979 / 2
x1 ≈ 44.95
x2 = (-400 - 489.8979) / 2
x2 = -889.8979 / 2
x2 ≈ -444.95
Therefore, the break-even points are approximately x = 44.95 and x = -444.95.
To find the break-even points, we need to determine the values of x where the total costs (C(x)) equal the total revenues (R(x)).
Setting C(x) equal to R(x), we have:
20,000 + 25x + 0.2x^2 = 475x − 0.8x^2.
Combining like terms:
0.2x^2 + 0.8x^2 - 475x + 25x - 20,000 = 0.
Simplifying further:
x^2 + 5x - 20,000 = 0.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
In this case, a = 1, b = 5, and c = -20,000.
Substituting these values into the quadratic formula:
x = (-5 ± √(5^2 - 4(1)(-20,000))) / (2(1)).
Calculating further:
x = (-5 ± √(25 + 80,000)) / 2.
x = (-5 ± √80,025) / 2.
We need to find the two possible values of x, so we consider both the positive and negative square root:
x1 = (-5 + √80,025) / 2.
x2 = (-5 - √80,025) / 2.
Evaluating the square root:
x1 = (-5 + 283.17) / 2.
x2 = (-5 - 283.17) / 2.
Simplifying further:
x1 = 278.17 / 2.
x2 = -288.17 / 2.
Calculating the final results:
x1 = 139.085.
x2 = -144.085.
Therefore, the break-even points for this company are approximately x = 139.085 and x = -144.085. Note that a negative value for x may not have practical significance in this context, so the break-even point is approximately x = 139.085.
break-even where R=C, so just solve
20,000 + 25x + 0.2x^2 = 475x − 0.8x^2
Rearrange and collect terms, and then just use the quadratic formula.