A certain fast food restaurant wants to determine the increase in revenue per cheeseburger if sales are $45,000. Let the function [p=(120000-x)/(30000)] represent the price of x cheeseburgers.
To determine the increase in revenue per cheeseburger, we need to calculate the derivative of the revenue function with respect to x. In this case, the revenue function is represented by the equation p = (120000 - x) / (30000), where p is the price of x cheeseburgers.
Let's start by rearranging the equation to find the revenue function R(x):
p = (120000 - x) / (30000)
Multiply both sides of the equation by 30000:
30000p = 120000 - x
Rearrange the equation to solve for x:
x = 120000 - 30000p
Now, let's express the revenue function in terms of x:
R(x) = x * p
Substitute the expression for x we just obtained:
R(x) = (120000 - 30000p) * p
Expand the equation:
R(x) = 120000p - 30000p^2
To find the increase in revenue per cheeseburger, we need to calculate the derivative of R(x) with respect to x, which is dR/dx.
dR/dx = d(120000p - 30000p^2)/dx
Differentiate each term with respect to x:
dR/dx = d(120000p)/dx - d(30000p^2)/dx
The derivative of a constant with respect to x is zero, so the first term becomes:
d(120000p)/dx = 120000 * dp/dx
To differentiate the second term, we need to use the chain rule:
d(30000p^2)/dx = 2 * 30000p * dp/dx
Now, combine the two terms:
dR/dx = 120000 * dp/dx - 60000p * dp/dx
Factor out dp/dx:
dR/dx = (120000 - 60000p) * dp/dx
To find the increase in revenue per cheeseburger, substitute the given sales figure of $45,000 into the equation and solve for dp/dx:
45000 = (120000 - 60000p) * dp/dx
Now, we can solve for dp/dx by rearranging the equation:
dp/dx = 45000 / (120000 - 60000p)
Therefore, the increase in revenue per cheeseburger can be calculated by finding dp/dx using the equation dp/dx = 45000 / (120000 - 60000p), where p represents the price of cheeseburgers.
To determine the increase in revenue per cheeseburger when sales are $45,000, we need to find the derivative of the revenue function with respect to the number of cheeseburgers sold (x).
Given the function p = (120000 - x) / 30000, where p represents the price of x cheeseburgers, we can first express the revenue function as R = p * x.
R = (120000 - x) / 30000 * x
Now, let's find the derivative of R with respect to x.
dR/dx = (d/dx)((120000 - x) / 30000 * x)
Using the product rule, the derivative of R with respect to x can be written as:
dR/dx = (1/30000) * x - (120000 - x) * 1 / 30000
Simplifying further:
dR/dx = x / 30000 - (120000 - x) / 30000
dR/dx = (2x - 120000) / 30000
Now, we can substitute the given sales value of $45,000 (x = 45,000) into the derivative to find the increase in revenue per cheeseburger:
dR/dx = (2 * 45,000 - 120,000) / 30,000
dR/dx = (90,000 - 120,000) / 30,000
dR/dx = -30,000 / 30,000
dR/dx = -1
Therefore, the increase in revenue per cheeseburger when sales are $45,000 is -1.
p is price of x burger, then divide by x to get price per burger
Priceburger= 1/x * p
= 120000/30000x -1/30000
dPriceburger/dx= -4x^2
put in x=45000, and you have the increase in revenue per burger. assuming costs are the same (which they never are).