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 find the derivative of the revenue function with respect to the number of cheeseburgers sold. The given function represents the price of x cheeseburgers.
Revenue is calculated by multiplying the number of cheeseburgers sold (x) by the price per cheeseburger (p):
Revenue = x * p
Now, let's differentiate the revenue function with respect to x:
dRevenue/dx = d(x * p)/dx
To find this derivative, we will use the product rule from calculus. The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:
d(uv)/dx = u * dv/dx + v * du/dx
In our case, u(x) = x and v(x) = p. The derivative of u with respect to x is du/dx = 1, as x is just a linear function. To find dv/dx, we need to differentiate p(x).
The given function for p(x) is p = (120000 - x) / 30000.
So, let's calculate dv/dx:
dv/dx = d((120000 - x) / 30000)/dx
To simplify this, let's rewrite p as (120000 - x) * (1/30000):
p = (120000 - x) * (1/30000)
Now, we can apply the quotient rule of differentiation. The quotient rule states that if we have two functions, u(x) and v(x), then the derivative of their quotient is given by:
d(u/v)/dx = (v * du/dx - u * dv/dx) / v^2
In our case, u(x) = (120000 - x) and v(x) = 30000. The derivative of u with respect to x is du/dx = -1, as x is linear. To calculate dv/dx, we need to differentiate v(x) = 30000:
dv/dx = d(30000)/dx = 0
Using the quotient rule, let's find dv/dx:
dv/dx = ((30000 * -1) - (120000 - x) * 0) / (30000)^2
= -30000 / (30000)^2
= -1 / 30000
Now that we have dv/dx, we can substitute it back into the original differentiation expression and calculate dRevenue/dx:
dRevenue/dx = x * (-1/30000) + p
Substituting the given price function back into the expression:
dRevenue/dx = x * (-1/30000) + (120000 - x) / 30000
Simplifying this expression will give us the derivative of the revenue function with respect to the number of cheeseburgers sold.