A pharmaceutical corporation has two locations that produce the same over-the-counter medicine. If x1 and x2 are the numbers of units produced at location 1 and location 2, respectively, then the total revenue for the product is given by
R = 700x1 + 700x2 − 2x1^2 − 4x1x2 − 2x2^2.
When x1 = 4 and x2 = 12, find the following.
(a) the marginal revenue for location 1,
∂R/∂x1
(b) the marginal revenue for location 2,
∂R/∂x2
∂R/∂x1 = 700 - 4x1 - 4x2
∂R/∂x2 = 700 - 4x1 - 4x2
To find the marginal revenue for location 1 (∂R/∂x1), we need to take the derivative of the total revenue function with respect to x1 while holding x2 constant.
First, let's find the derivative of each term separately:
d(700x1)/dx1 = 700 (since x1 is the only term involving x1)
d(-2x1^2)/dx1 = -4x1 (using the power rule, we reduce the exponent by 1 and multiply by the original exponent)
d(-4x1x2)/dx1 = -4x2 (since x1 is constant, we treat x2 as a constant and differentiate it)
d(-2x2^2)/dx1 = 0 (since x1 is constant, we treat x2 as a constant and differentiate it, resulting in zero)
Now, we can sum up these derivatives to find the marginal revenue (∂R/∂x1):
∂R/∂x1 = 700 - 4x1 - 4x2
Plugging in the values x1 = 4 and x2 = 12:
∂R/∂x1 = 700 - 4(4) - 4(12) = 700 - 16 - 48 = 636
Therefore, the marginal revenue (∂R/∂x1) for location 1 is 636.
To find the marginal revenue for location 2 (∂R/∂x2), we need to take the derivative of the total revenue function with respect to x2 while holding x1 constant.
Following the same steps as above, we find:
∂R/∂x2 = 700 - 2x2 - 4x1
Plugging in the values x1 = 4 and x2 = 12:
∂R/∂x2 = 700 - 2(12) - 4(4) = 700 - 24 - 16 = 660
Therefore, the marginal revenue (∂R/∂x2) for location 2 is 660.