Search 72,401 tutors FIND TUTORS
Search for tutors
Ask a question
0 0

Help solving this three-part word problem?

At $0.49 per bushel, the daily supply for wheat is 305 bushels, and the daily demand it 515 bushels.  When the price is raised to $0.73 per bushel, the daily supply increases to 545 bushels, and the daily demand decreases to 435 bushels.  Assume that the price-supply and price-demand equations are linear.
 
A.) Find the price-supply equation
B.) Find the price-demand equation
C.) Find the equilibrium price and equilibrium quantity
 
I'm particularly having trouble with part C. but if you could work this entire problem out for me I would appreciate it!
Tutors, please sign in to answer this question.

2 Answers

A.)  Since the price supply equation is linear, it will have the general form
 
       y = ax + b 
 
Let y = the number of bushels and x the price per bushel:
 
       y = a($0.73) + b ⇒ 545 bushels = ($0.73)a + b
       y = a($0.49) + b ⇒ 305 bushels = ($0.49)a + b
       
 
       Subtract the two equations
 
        (545 bushels - 305 bushels) = a($0.73 - $0.49)
 
       240 bushels = a($0.24)
 
       ∴ a = (240 bushels)/($.24) = 1000 bushels per $
 
        To find the value of b, plug the value of a into either of the original equations:
 
        y = 545 = (1000)($0.73) + b
 
        b = 545 - 730 = -185
 
       So the price-supply equation may be written as: y = (1000)x - 185
 
B.)   For the price-demand equation use the same procedure
 
       y = a($0.73) + b ⇒ 435 bushels = ($0.73)a + b
       y = a($0.49) + b ⇒ 515 bushels = ($0.49)a + b
 
       (435 bushels - 515 bushels) = a($0.73 - $0.49)
 
       -80 bushels = ($0.24)a
 
       a = (-80 bushels)/($0.24) = -333.33
 
       435 bushels = ($0.73)(-333.33) + b
 
       b = 435 bushels - ($0.73)(-333.33) = 678.33
 
       The price-demand equation is: y = -333.33x + 683.33
 
C.)   The equilibrium price and quantity are at the point where these two lines intersect.
 
       y = (1000)x - 185
       y = -333.33x + 683.33
 
       1000x - 185 = -333.333x + 683.333
 
       x(1000 + 333.333) = 683.333 + 185
 
       1333.333*x = 868.333
 
       x = (683.333)/(1333.333) = $0.65 (equilibrium)
 
       y = (1000)($0.65) - 185 = 466.25 bushels (equilibrium)
 
       
 
      

Comments

Courtnee,
 
I should have mentioned yesterday that we could have just as easily allowed the number of bushels to be the independent variable x and the price per bushel be the dependent variable y.  That's one of the nice things about linear equations.

Comment

A.) Points for price-supply, (p,s): ($0.49/b, 305 b), ($0.73/b, 545 b)
 
s(p) - 305 = ((545-305)/(0.73-0.49))(p - 0.49)
s(p) - 305 = (240/(0.24))(p - 0.49)
s(p)  = 1000(p - 0.49) + 305
s(p) = 1000p - 490 + 305
s(p) = 1000p - 185 b
 
B.) Points for price-demand, (p,d): ($0.49/b, 515 b), ($0.73/b, 435 b)
 
d(p) - 515 = ((515-435)/(0.49-0.73))(p - 0.49)
d(p) - 515 = (80/(-0.24))(p - 0.49)
d(p) = -(1000/3)(p - 0.49) + 515
d(p) = -1000p/3 + 490/3 + 1545/3
d(p) = -1000p/3 + 2035/3 b
 
C.) Use Substitution:
1000p - 185 = -1000p/3 + 2035/3
Multiply by 3:
3000p - 555 = -1000p + 2035
4000p = 555 + 2035 = 2590
p = 259/400 = 2.59/4 = $0.6475/b
 
s($0.6475/b) = d($0.6475/b) = 1000(0.6475) - 185 = 647.5 -185 = 462.5 b

Comments

Note that answers are exact.
Thank you both for your help!  I better understand the process but really the only problem is how the answers are input.  The correct answers are:
 
A.) p=0.001q+0.185
 
B.) p=-0.003q+2.035
 
C.) Price=$.65
     Quantity=463 bushels
 
But thanks again for your time!
-Courtnee

Comment