If a firm has the following cost anfd revenue functions find the break even points
C(x)=3600+25x+1/2x^2
R(x)=(175-1/2x)x
I hope you mean:
C(x)=3600+25x+(1/2)x^2
R(x)=(175-(1/2)x)x
where does C = R?
.5 x^2 + 25 x +3600 = 175 x - .5 x^2
x^2 -150 x + 3600 = 0
(x-120)(x-30) = 0
x = 120 or x = 30
To find the break-even points, we need to determine the values of x where the revenue (R(x)) equals the cost (C(x)).
First, let's set up the equation:
C(x) = R(x)
Substituting the given cost and revenue functions:
3600 + 25x + 1/2x^2 = (175 - 1/2x)x
Let's simplify the equation by combining like terms and rearranging:
1/2x^2 + 25x + 3600 = 175x - 1/2x^2
Adding 1/2x^2 to both sides:
x^2 + 50x + 7200 = 350x
Rearranging the equation:
x^2 - 300x + 7200 = 0
Now, we can solve this quadratic equation.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here:
a = 1
b = -300
c = 7200
Substituting these values into the quadratic formula:
x = (-(-300) ± √((-300)^2 - 4(1)(7200))) / (2(1))
x = (300 ± √(90000 - 28800)) / 2
x = (300 ± √(61200)) / 2
x = (300 ± 247.49) / 2
Therefore, the break-even points are:
x1 = (300 + 247.49) / 2 = 273.75
x2 = (300 - 247.49) / 2 = 26.25
So, the break-even points for the given cost and revenue functions are x = 273.75 and x = 26.25.
To find the break-even points, we need to determine the values of x where the cost (C(x)) is equal to the revenue (R(x)). In other words, we want to find the quantity level x where the firm neither makes a profit nor incurs a loss.
Step 1: Set C(x) equal to R(x) and solve for x:
3600 + 25x + 1/2x^2 = (175 - 1/2x)x
Step 2: Simplify the equation by expanding and collecting like terms:
3600 + 25x + 1/2x^2 = 175x - 1/2x^2
Rearranging the terms:
1/2x^2 + 1/2x^2 - 25x + 175x - 3600 = 0
Combine like terms:
x^2 + 150x - 3600 = 0
Step 3: Solve the quadratic equation. There are multiple ways to solve this equation, but let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
With a = 1, b = 150, and c = -3600, we can substitute these values into the quadratic formula:
x = (-150 ± √(150^2 - 4(1)(-3600))) / (2(1))
Simplifying further:
x = (-150 ± √(22500 + 14400)) / 2
x = (-150 ± √36900) / 2
Step 4: Calculate the two potential break-even points by evaluating both the positive and negative square root:
x1 = (-150 + √36900) / 2
x2 = (-150 - √36900) / 2
After performing the calculations, these values will give you the two break-even points for the firm.