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You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize its weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form q = mp + b, where p is the price per hamburger, q is the demand in weekly sales, and m and b are certain constants you must determine.
(a) Your market studies reveal the following sales figures: When the price is set at $2.00 per hamburger, the sales amount to 6000 per week, but when the price is set at $4.00 per hamburger, the sales drop to zero. Use these data to calculate the demand equation.
(b) Now estimate the unit price that maximizes weekly revenue.
Predict what the weekly revenue will be at that price.
so, you have two points: (2,6000) and (4,0)
Use them to find the values of m and b...
To determine the demand equation, we will use the given sales figures at two different prices. Let's start with part (a) by calculating the demand equation using the given data points.
(a) To calculate the demand equation, we need to find the values of constants 'm' and 'b' in the linear equation q = mp + b.
Let's consider the first data point:
When the price is set at $2.00 per hamburger, the sales amount to 6000 per week.
Using this information, we can plug in the values into the demand equation:
6000 = 2m + b (Equation 1)
Now, let's consider the second data point:
When the price is set at $4.00 per hamburger, the sales drop to zero.
Using this information, we can plug in the values into the demand equation:
0 = 4m + b (Equation 2)
Now, we have a system of equations with two unknowns (m and b). We can solve this system of equations to find the values of m and b.
To solve the system of equations, we can subtract Equation 2 from Equation 1:
(6000 - 0) = (2m + b) - (4m + b)
6000 = -2m
Dividing both sides by -2, we get:
m = -3000
Now, let's substitute the value of m into either Equation 1 or Equation 2 to find the value of b. Let's use Equation 1:
6000 = 2(-3000) + b
6000 = -6000 + b
12000 = b
So, we have found the values of m and b for the demand equation:
m = -3000
b = 12000
Therefore, the demand equation is q = -3000p + 12000.
Now, let's move to part (b) and estimate the unit price that maximizes the weekly revenue.
(b) To maximize the weekly revenue, we need to find the price at which the demand equation gives the highest value of weekly revenue.
The revenue equation (R) is given by multiplying the demand equation (q) by the price (p):
R = pq = p(-3000p + 12000) = -3000p^2 + 12000p
To find the price that maximizes weekly revenue, we can take the derivative of the revenue equation with respect to price (p) and set it to zero. Then, solve for p.
dR/dp = -6000p + 12000
Setting dR/dp to zero:
-6000p + 12000 = 0
Dividing both sides by -6000, we get:
p = 2
Therefore, the unit price that maximizes the weekly revenue is $2.
To predict the weekly revenue at that price, substitute p = 2 back into the revenue equation:
R = -3000(2)^2 + 12000(2)
R = -12000 + 24000
R = 12000
The predicted weekly revenue at the unit price of $2 is $12,000.