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.

Question

p equals 106.60 minus 0.10 q ​(demand function)
p equals 0.01 q squared plus 4.84 ​(supply function)
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice.
A.
The equilibrium price of ​$
  
enter your response here gives a demand that is equal to a supply of
  
enter your response here hundred trees.
​(Simplify your answer. Type integers or simplified​ fractions.)
B.
The equilibrium price does not exist.

Answer 1

To find the market equilibrium price, we need to set the demand function equal to the supply function and solve for \( p \) and \( q \).

The demand function is given by: \[ p = 106.60 - 0.10q \]

The supply function is given by: \[ p = 0.01q^2 + 4.84 \]

Setting these two functions equal to each other gives us: \[ 106.60 - 0.10q = 0.01q^2 + 4.84 \]

Now let's rearrange the equation: \[ 0.01q^2 + 0.10q + 4.84 - 106.60 = 0 \]

This simplifies to: \[ 0.01q^2 + 0.10q - 101.76 = 0 \]

Next, to eliminate the decimal, we can multiply the entire equation by 100: \[ q^2 + 10q - 10176 = 0 \]

Now we can use the quadratic formula to solve for \( q \): \[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 10 \), and \( c = -10176 \).

Calculating the discriminant: \[ b^2 - 4ac = 10^2 - 4(1)(-10176) = 100 + 40604 = 40704 \]

Now we can find \( q \): \[ q = \frac{-10 \pm \sqrt{40704}}{2(1)} \] Calculating \( \sqrt{40704} \): \[ \sqrt{40704} \approx 202 \] So, \[ q = \frac{-10 \pm 202}{2} \]

We will only consider the positive root since quantity cannot be negative: \[ q = \frac{-10 + 202}{2} = \frac{192}{2} = 96 \]

Now that we have \( q = 96 \), we can substitute this back into either the demand or supply function to find the equilibrium price \( p \).

Using the demand function: \[ p = 106.60 - 0.10(96) = 106.60 - 9.60 = 97 \]

Thus, the equilibrium price is: \[ \text{Equilibrium price: } p = 97 \text{ dollars} \]

And the equilibrium quantity (in hundreds of trees) is: \[ \text{Quantity: } q = 96 \text{ (or } 9600 \text{ trees)} \]

Therefore, the final answers are: A. The equilibrium price of $97 gives a demand that is equal to a supply of 96 hundred trees.