Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week of its rolltop desks in the finished and unfinished versions are x and y units when the corresponding unit prices, in dollars, are described, respectively, in the following equations.
p = 310-1x-1/2y
q = 730-1/2x-1/7y
(a) What is the weekly total revenue function R(x,y)?
NOTE; Someone else gave me this answer but it still says it is wrong. Thanks
310x - 1x^2 - 1/2xy + 750y -1/2xy -1/7y^2 -1x^2 -1/7y^2 + 310x -xy + 750y
Even if the above formula is correct, it would surely not be the correct answer, since like terms have not been combined, and the order of powers is unconventional. Also, there is no constant term, even though both P and Q contain a constant term, so there is no way it could be correct.
revenue = price*quantity, so
R(x,y) = P(x,y)*Q(x,y)
1/2 x^2 + 11/28 xy + 1/14 y^2 - 885x - 2865/7 y + 226300
I leave it to you to verify it.
To find the weekly total revenue function, we need to multiply the quantity demanded of each product by its corresponding unit price and then sum them up.
Given that the unit price of the finished rolltop desks is described by p = 310 - x - 1/2y and the unit price of the unfinished rolltop desks is described by q = 730 - 1/2x - 1/7y, we can express the total revenue as follows:
Total revenue = (Unit price of finished rolltop desks * Quantity demanded of finished rolltop desks) + (Unit price of unfinished rolltop desks * Quantity demanded of unfinished rolltop desks)
Substituting the given equations for the unit prices, we have:
Total revenue = (310 - x - 1/2y) * x + (730 - 1/2x - 1/7y) * y
Expanding and simplifying this expression, we get:
Total revenue = 310x - x^2 - 1/2xy + 730y - 1/2xy - 1/7y^2
Combining like terms, we can simplify further:
Total revenue = 310x - x^2 - xy - 1/7y^2 + 730y
Hence, the correct weekly total revenue function R(x,y) is:
R(x,y) = 310x - x^2 - xy - 1/7y^2 + 730y
To find the weekly total revenue function R(x,y), we need to multiply the quantity demanded of each version of the furniture by its corresponding unit price.
Given the unit prices:
p = 310 - x - 1/2y
q = 730 - 1/2x - 1/7y
To find the total revenue for the finished furniture (p), we multiply the quantity demanded (x) by the unit price (p):
Revenue from finished furniture (Rf) = x * p = x(310 - x - 1/2y)
To find the total revenue for the unfinished furniture (q), we multiply the quantity demanded (y) by the unit price (q):
Revenue from unfinished furniture (Ru) = y * q = y(730 - 1/2x - 1/7y)
The weekly total revenue function R(x,y) is the sum of the revenues from finished and unfinished furniture:
R(x,y) = Rf + Ru
= x(310 - x - 1/2y) + y(730 - 1/2x - 1/7y)
Expanding and simplifying:
R(x,y) = 310x - x^2 - 1/2xy + 730y - 1/2xy - 1/7y^2
= 310x - x^2 - xy + 730y - 1/7y^2
Therefore, the weekly total revenue function is:
R(x,y) = 310x - x^2 - xy + 730y - 1/7y^2