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 160 guests per night. The next week you charged $11 per guest and averaged 140 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.
q = ap+b
160 = 9a+b
140 = 11a+b
a = -10
b = 250
q = 250 - 10p
To find a linear demand equation showing the number of guests per night based on the cover charge, we need to use the concept of slope-intercept form of a linear equation, which is y = mx + b, where y represents the number of guests per night, x represents the cover charge, m represents the slope, and b represents the y-intercept.
In this case, we can use the data from the two weeks to find the slope and y-intercept. Let's use the first week as an example:
Week 1:
Cover charge (x) = $9
Number of guests (y) = 160
We can substitute these values into the slope-intercept form equation and solve for b:
160 = m * 9 + b (equation 1)
Next, let's use the data from the second week:
Week 2:
Cover charge (x) = $11
Number of guests (y) = 140
Again, we can substitute these values into the equation and solve for b:
140 = m * 11 + b (equation 2)
Now, we have a system of two equations with two variables (m and b). We can solve this system of equations to find the values of m and b, which will give us the linear demand equation. Here's how:
Step 1: Solve equation 2 for b in terms of m:
b = 140 - m * 11
Step 2: Substitute this value of b into equation 1:
160 = m * 9 + (140 - m * 11)
Simplify the equation:
160 = 9m + 140 - 11m
Combine like terms:
160 = -2m + 140
Subtract 140 from both sides:
20 = -2m
Divide by -2:
m = -10
Step 3: Substitute this value of m back into equation 2 to solve for b:
140 = -10 * 11 + b
Simplify the equation:
140 = -110 + b
Add 110 to both sides:
b = 250
Therefore, the linear demand equation is:
q = -10p + 250
where q represents the number of guests per night and p represents the cover charge.