You are given a pair of equations, one representing a supply curve and the other representing a demand curve, where p is the unit price for x items.
−p+0.0208333333333333x+2=0
and
p = √(57-x)
What is the market equilibrium for x?
I know I'm supposed to set both equations equal to each other, but I'm having trouble solving for x because of the radical sign.
Eh? Remember your Algebra I?
-√(57-x) + 0.02083x + 2 = 0
-√(57-x) = -0.02083x - 2
57-x = (-0.02083x - 2)^2
now just expand, solve for x as usual, and watch out for extraneous solutions.
I tried and got 3.371880575, but I think I'm still wrong...
To find the market equilibrium for x, we need to set the supply curve equal to the demand curve and solve for x.
First, let's start by setting the two equations equal to each other:
−p + 0.0208333333333333x + 2 = p
Next, let's replace p in the equation using the second equation:
−√(57-x) + 0.0208333333333333x + 2 = √(57-x)
Now, we can simplify the equation by moving all terms to the left side:
−√(57-x) - √(57-x) + 0.0208333333333333x + 2 = 0
Simplifying further:
2√(57-x) + 0.0208333333333333x - 2 = 0
Now, let's solve for x.
1. Move the constant term to the right side:
2√(57-x) + 0.0208333333333333x = 2
2. Square both sides of the equation to eliminate the radical:
(2√(57-x))^2 + (0.0208333333333333x)^2 + 2(2√(57-x))(0.0208333333333333x) = 2^2
4(57-x) + 0.00043402777777777x^2 + 0.16668981502777782x√(57-x) = 4
3. Simplify the equation:
228 - 4x + 0.00043402777777777x^2 + 0.16668981502777782x√(57-x) = 4
4. Move all the terms to one side of the equation:
0.00043402777777777x^2 + 0.16668981502777782x√(57-x) - 4x + 224 = 0
Now, we have a quadratic equation in terms of x that we can solve using numerical methods, such as graphing or a quadratic formula solver. Unfortunately, we can't provide the exact solution for x without using additional tools.
Once you find the solutions for x, you can substitute them back into either the supply or demand equation to determine the corresponding unit price (p) at market equilibrium.
Keep in mind that the equations provided may have multiple solutions or complex solutions, so it's essential to check the validity of any obtained answers in the context of the problem.