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)?
(b) Find the domain of the function R(x,y).
p= ?
q=?
p = 310-1x-1/2y
q = 730-1/2x-1/7y
(a) What is the weekly total revenue function R(x,y) = xp + yq
310x - 1x^2 - 1/2xy + 750y -1/2xy -1/7y^2
-1x^2 -1/7y^2 + 310x -xy + 750y
(b) Find the domain of the function R(x,y).
p= ? x=>0
q=? y =>0
-1x^2 -1/7y^2 + 310x -xy + 730y
(a) The weekly total revenue function, R(x,y), can be calculated by multiplying the quantity demanded (q) by the unit price (p):
R(x,y) = p*q
Given that p = 310 - x - 1/2y and q = 730 - 1/2x - 1/7y, we can substitute these values into the equation to find the total revenue function:
R(x,y) = (310 - x - 1/2y) * (730 - 1/2x - 1/7y)
Simplifying further:
R(x,y) = 226300 - 310x - 155y - 365/2x + xy/2 + 35/7y + xy/14
Combining like terms:
R(x,y) = 226300 - 437x - 120/7y + xy/2 + xy/14
Therefore, the weekly total revenue function is R(x,y) = 226300 - 437x - 120/7y + xy/2 + xy/14.
(b) To find the domain of the function R(x,y), we need to consider any restrictions or limitations on the variables x and y. In this case, the domain will be all possible values of x and y that make the equation valid.
Looking at the given equations for p and q, two restrictions can be identified: x ≥ 0 and y ≥ 0. This is because quantities cannot be negative in the real world.
Therefore, the domain of the function R(x,y) is x ≥ 0 and y ≥ 0.
To find the values of p and q separately, substitute x and y into their respective equations:
p = 310 - x - 1/2y
q = 730 - 1/2x - 1/7y