`S(P) =1/2P + 40`
`D(P) = 220-P`
The quantity of a product supplied and the quantity of the product demanded in an economic market are functions of the price of the product. The functions above are the estimated supply and demand functions for a certain product. The function S P( ) gives the quantity of the product supplied to the market when the price is P dollars, and the function D P( ) gives the quantity of the product demanded by the market when the price is P dollars.
At what price will the quantity of the product supplied to the market equal the quantity of the product demanded by the market?

Options

A. $ 90
B. $120
C. $133
D. $155

Answer

To find the price at which the quantity of the product supplied equals the quantity demanded, we need to set the supply function \( S(P) \) equal to the demand function \( D(P) \).

1. **Write down the supply and demand functions:**
\[ S(P) = \frac{1}{2}P + 40 \]
\[ D(P) = 220 - P \]

2. **Set the supply function equal to the demand function:**
\[ \frac{1}{2}P + 40 = 220 - P \]

3. **Rearrange the equation to isolate \( P \):**
- First, add \( P \) to both sides:
\[ \frac{1}{2}P + P + 40 = 220 \]
- This simplifies to:
\[ \frac{3}{2}P + 40 = 220 \]

4. **Subtract 40 from both sides:**
\[ \frac{3}{2}P = 220 - 40 \]
\[ \frac{3}{2}P = 180 \]

5. **Multiply both sides by \( \frac{2}{3} \) to solve for \( P \):**
\[ P = \frac{2}{3} \times 180 \]
- Simplifying this gives:
\[ P = 120 \]

6. **Conclusion:**
The price at which the quantity supplied equals the quantity demanded is \( P = 120 \) dollars.