To find the demand function, we start with the information given. We know that the slope of the demand function is [tex]\(-\frac{1}{2}\)[/tex] and we have a specific point on this line: when the quantity demanded is 10 units, the price is [tex]$50.
### Step 1: Calculate the Y-Intercept (Price Intercept)
The general form of the linear equation is:
\[ \text{price} = \text{slope} \times \text{quantity} + \text{intercept} \]
Given the slope \( -\frac{1}{2} \) and a point \((10, 50)\), we can substitute these values into the equation to find the intercept.
\[ 50 = -\frac{1}{2} \times 10 + \text{intercept} \]
Simplifying, we get:
\[ 50 = -5 + \text{intercept} \]
Adding 5 to both sides to solve for the intercept:
\[ \text{intercept} = 50 + 5 = 55 \]
So, the y-intercept of the demand function is 55.
### Step 2: Form the Demand Function
Now that we know the slope and the y-intercept, we can write the demand function as:
\[ \text{price} = -\frac{1}{2} \times \text{quantity} + 55 \]
### Step 3: Calculate the X-Intercept (Quantity Intercept)
The x-intercept occurs where the price is 0. We set the price to 0 in the demand function and solve for the quantity.
\[ 0 = -\frac{1}{2} \times \text{quantity} + 55 \]
Multiplying through by 2 to clear the fraction:
\[ 0 = -1 \times \text{quantity} + 110 \]
Solving for the quantity:
\[ \text{quantity} = 110 \]
So, the x-intercept of the demand function is 110.
### Step 4: Calculate the Price at a Quantity of 50 Units
We use the demand function to find the price when the quantity demanded is 50 units.
\[ \text{price} = -\frac{1}{2} \times 50 + 55 \]
Simplifying, we get:
\[ \text{price} = -25 + 55 = 30 \]
So, the price when the quantity demanded is 50 units is $[/tex]30.
### Summary
- The y-intercept (price intercept) is [tex]\( \$55 \)[/tex].
- The x-intercept (quantity intercept) is [tex]\( 110 \)[/tex] units.
- The price when the quantity demanded is 50 units is [tex]\( \$30 \)[/tex].
Hence, the demand function is:
[tex]\[ \text{price} = -\frac{1}{2} \times \text{quantity} + 55 \][/tex]
These intercepts and the price at 50 units demonstrate the linear relationship described by the demand function.
