SOLUTION: The demand equation for a certain type of printer is given by: D = -200p + 35,000 The supply equation is predicted to be S = -p^2 + 400p - 20,000 Find the equilibrium price.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: The demand equation for a certain type of printer is given by: D = -200p + 35,000 The supply equation is predicted to be S = -p^2 + 400p - 20,000 Find the equilibrium price.       Log On


   

Question 90296This question is from textbook Beginning Al
: The demand equation for a certain type of printer is given by:
D = -200p + 35,000
The supply equation is predicted to be
S = -p^2 + 400p - 20,000
Find the equilibrium price.
No clue on what to do...help is appreciated! This question is from textbook Beginning Al

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To find the equilibrium price, set D equal to S
-200p%2B35000=-p%5E2+%2B+400p+-+20000

-200p%2B35000%2Bp%5E2=400p-20000 Add p%5E2 to both sides

-200p%2B35000%2Bp%5E2-400p=-20000Subtract 400p from both sides

-200p%2B35000%2Bp%5E2-400p%2B20000=0 Add 20000 to both sides

p%5E2-600p%2B55000=0 Combine like terms



Now let's use the quadratic formula to solve for p:


Starting with the general quadratic

ap%5E2%2Bbp%2Bc=0

the general solution using the quadratic equation is:

p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve p%5E2-600%2Ap%2B55000=0 ( notice a=1, b=-600, and c=55000)

p+=+%28--600+%2B-+sqrt%28+%28-600%29%5E2-4%2A1%2A55000+%29%29%2F%282%2A1%29 Plug in a=1, b=-600, and c=55000



p+=+%28600+%2B-+sqrt%28+%28-600%29%5E2-4%2A1%2A55000+%29%29%2F%282%2A1%29 Negate -600 to get 600



p+=+%28600+%2B-+sqrt%28+360000-4%2A1%2A55000+%29%29%2F%282%2A1%29 Square -600 to get 360000



p+=+%28600+%2B-+sqrt%28+360000%2B-220000+%29%29%2F%282%2A1%29 Multiply -4%2A55000%2A1 to get -220000



p+=+%28600+%2B-+sqrt%28+140000+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



p+=+%28600+%2B-+100%2Asqrt%2814%29%29%2F%282%2A1%29 Simplify the square root



p+=+%28600+%2B-+100%2Asqrt%2814%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

p+=+%28600+%2B+100%2Asqrt%2814%29%29%2F2 or p+=+%28600+-+100%2Asqrt%2814%29%29%2F2


which split up to


p=%2B600%2F2%2B100%2Asqrt%2814%29%2F2 or p=%2B600%2F2-100%2Asqrt%2814%29%2F2


and simplify to


p=300%2B50%2Asqrt%2814%29 or p=300-50%2Asqrt%2814%29


Which approximate to

p=487.082869338697 or p=112.917130661303


So our solutions are:
p=487.082869338697 or p=112.917130661303



However, since the first solution x=487.082869338697 produces a negative demand value, we must discard the first solution.

So the equilibrium price is $112.92