Questions: vork 4.1 Question 7, 4.1.41 Part 3 of 4 HW Score: 58%, 5.8 of 10 poin Points: 0 of 1 Suppose that the quantity supplied S and quantity demanded D of T-shirts at a concert are given by the following functions where p is the price. S(p)=-200+40p D(p)=1150-50p Answer parts (a) through (c). (a) Find the equilibrium price for the T-shirts at this concert. The equilibrium price is 15 (Round to the nearest dollar as needed.) What is the equilibrium quantity? The equilibrium quantity is 400 T-shirts. (Type a whole number.) (b) Determine the prices for which quantity demanded is greater than quantity supplied. For the price , the quantity demanded is greater than quantity supplied.

vork 4.1 Question 7, 4.1.41 Part 3 of 4 HW Score: 58%, 5.8 of 10 poin Points: 0 of 1 Suppose that the quantity supplied S and quantity demanded D of T-shirts at a concert are given by the following functions where p is the price. S(p)=-200+40p D(p)=1150-50p Answer parts (a) through (c). (a) Find the equilibrium price for the T-shirts at this concert. The equilibrium price is 15 (Round to the nearest dollar as needed.) What is the equilibrium quantity? The equilibrium quantity is 400 T-shirts. (Type a whole number.) (b) Determine the prices for which quantity demanded is greater than quantity supplied. For the price , the quantity demanded is greater than quantity supplied.
Transcript text: vork 4.1 Question 7, 4.1.41 Part 3 of 4 HW Score: 58\%, 5.8 of 10 poin Points: 0 of 1 Suppose that the quantity supplied $S$ and quantity demanded $D$ of T-shirts at a concert are given by the following functions where $p$ is the price. \[ \begin{array}{l} S(p)=-200+40 p \\ D(p)=1150-50 p \end{array} \] Answer parts (a) through (c). (a) Find the equilibrium price for the T-shirts at this concert. The equilibrium price is $\$ 15$ (Round to the nearest dollar as needed.) What is the equilibrium quantity? The equilibrium quantity is 400 T -shirts. (Type a whole number.) (b) Determine the prices for which quantity demanded is greater than quantity supplied. For the price $\$$ $\square$ \$ $\square$ , the quantity demanded is greater than quantity supplied.
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Solution

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Solution Steps

To solve the given problem, we need to find the equilibrium price and quantity where the quantity supplied equals the quantity demanded. Then, we determine the price range where the quantity demanded is greater than the quantity supplied.

Part (a)
  1. Set the supply function \( S(p) \) equal to the demand function \( D(p) \) to find the equilibrium price.
  2. Solve for \( p \) to find the equilibrium price.
  3. Substitute the equilibrium price back into either the supply or demand function to find the equilibrium quantity.
Part (b)
  1. Determine the inequality where the demand function \( D(p) \) is greater than the supply function \( S(p) \).
  2. Solve the inequality to find the range of prices where the quantity demanded is greater than the quantity supplied.
Step 1: Find the Equilibrium Price

To find the equilibrium price, we set the quantity supplied \( S(p) \) equal to the quantity demanded \( D(p) \): \[ S(p) = D(p) \] Substituting the given functions: \[ -200 + 40p = 1150 - 50p \] Solving for \( p \), we find: \[ 90p = 1350 \implies p = 15 \]

Step 2: Find the Equilibrium Quantity

Next, we substitute the equilibrium price \( p = 15 \) back into either the supply or demand function to find the equilibrium quantity: \[ S(15) = -200 + 40(15) = -200 + 600 = 400 \]

Step 3: Determine Price Range for Demand Greater than Supply

To find the prices for which the quantity demanded is greater than the quantity supplied, we solve the inequality: \[ D(p) > S(p) \] This leads to: \[ 1150 - 50p > -200 + 40p \] Solving this inequality gives: \[ -90p > -1350 \implies p < 15 \] Thus, the quantity demanded is greater than the quantity supplied for prices in the range: \[ (-\infty, 15) \]

Final Answer

The equilibrium price is \( \boxed{p = 15} \), the equilibrium quantity is \( \boxed{400} \), and the price range for which quantity demanded is greater than quantity supplied is \( \boxed{(-\infty, 15)} \).

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