Suppose that the demand and supply for artificial Christmas trees is given by the functions below where p is the price of a tree in dollars and q is the quantity of trees that are demanded/supplied in hundreds. Find the price that gives the market equilibrium price and the number of trees that will be sold/bought at this price.

P= 106.20-0.20q (demand function)

P= 0.01q^2+5.19 (supply function)

Question: The equilibrium price of $______ gives a demand that is equal to a supply of _______ hundred trees

Answers: The equilibrium price of a tree is $88 gives a demand that is equal to a supply of 91 hundred trees. 

find all the explanations for the answers.

Step-by-step explanation

Explanations: At the market equilibrium price, price of trees in dollar from demand function and price of trees in dollar from supply function would be same:

0.01q2+5.19=106.20−0.20q

0.01q2+0.20q+5.19−106.20=0

multiply above equation with 100:

q2+20q+519−10620=0

q2+20q−10101=0

try to find the factors of 10101, whose difference is equal to +20q and multiplication will be equal to −10101q2. so by observation, we have found that 111 and 91 are the two numbers, who fulfill the criteria. hence after putting these numbers in the above equation:

q2+111q−91q−10101=0

q(q+111)−91(q+111)=0

(q+111)(q−91)=0

q=−111a,q=91

as q is a number of trees, it can not be negative. hence the number of trees is 91 hundred. to find the equilibrium price, put the value of q=91 in the given equation:

peq?=106.20−0.20×91

peq?=106.20−81.2

peq?=$88

hence the equilibrium price of a tree is $88 gives a demand that is equal to a supply of 91 hundred trees.