Question
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 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.
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.
Answers
GPT-4o mini
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.
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.