You have just opened a new nightclub, Russ' Techno Pitstop, but are unsure of how high to set the cover charge (entrance fee). One week you charged $9 per guest and averaged 165 guests per night. The next week you charged $13 per guest and averaged 105 guests per night.
(a)
Find a linear demand equation showing the number of guests q per night as a function of the cover charge p.

To find a linear demand equation for the number of guests $ q $ per night as a function of the cover charge $ p $, we can use the two given points from the problem. The data points are:

- When the cover charge was \$9, there were 165 guests: $ (p_1, q_1) = (9, 165) $
- When the cover charge was \$13, there were 105 guests: $ (p_2, q_2) = (13, 105) $

We assume the relationship between the number of guests and the price is linear, meaning the equation will take the form:

\[
q = mp + b
\]

Where:
- $ m $ is the slope of the line,
- $ b $ is the y-intercept.

### Step 1: Find the slope $ m $
The slope of the line between two points $(p_1, q_1)$ and $(p_2, q_2)$ is calculated by:

\[
m = \frac{q_2 - q_1}{p_2 - p_1}
\]

Substitute the given values:

\[
m = \frac{105 - 165}{13 - 9} = \frac{-60}{4} = -15
\]

### Step 2: Find the y-intercept $ b $
Now that we know the slope is $ m = -15 $, we use one of the points to solve for $ b $. Let's use $ (9, 165) $:

\[
165 = -15(9) + b
\]

Simplifying:

\[
165 = -135 + b
\]

\[
b = 165 + 135 = 300
\]

### Step 3: Write the demand equation
Now that we have $ m = -15 $ and $ b = 300 $, the demand equation is:

\[
q = -15p + 300
\]

This is the linear demand equation showing the number of guests $ q $ per night as a function of the cover charge $ p $.

Would you like more details or have any questions? Here are some additional related questions to explore:

1. What does the slope of the demand equation represent?
2. What would happen to the number of guests if the cover charge increased to $15?
3. How many guests can you expect if you set the cover charge to $7?
4. What cover charge would attract exactly 150 guests?
5. How would the equation change if you lowered the cover charge to attract more guests?

**Tip:** When interpreting a linear demand equation, the slope tells you how much the demand (number of guests) decreases per dollar increase in the cover charge.

You have just opened a new nightclub, Russ' Techno Pitstop, but are unsure of how high to set the cover charge (entrance fee). One week you charged $9 per guest and averaged 165 guests per night. The next week you charged $13 per guest and averaged 105 guests per night. (a) Find a linear demand equation showing the number of guests q per night as a function of the cover charge p.

Learn how to calculate the linear demand equation for a nightclub based on cover charges. The problem uses algebraic concepts such as the slope formula and linear equations. Discover how changing the cover charge impacts the number of guests per night. Suitable for high school students in Grades 9-12.

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