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.

Question

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 = 200-1/5x-1/10y

q = 15-1/10x-1/4y
For the weekly total revenue function R(x, y), compute the values of R listed below using the given unit prices.
R(110, 69) = ?
R(61, 116) = ?

Answer 1

R(x , y) = 200x -1/5x^2 -1/10xy+ 15y - 1/10xy -1/4y^2

200x -1/5x^2 -1/5xy + 15y-1/4y^2

R(110, 69) = ? 17906.75

R(61, 116) = ?8416.60

Answer 2

To find the total revenue function R(x, y), we need to multiply the quantity demanded with the corresponding unit price.

Given unit prices:
p = 200 - 1/5x - 1/10y
q = 15 - 1/10x - 1/4y

To find R(x, y), we multiply p by x and q by y and add them together.

R(x, y) = px + qy

Now, let's calculate the values of R for the given quantities:

1. R(110, 69):
Substituting the values of x = 110 and y = 69 into the unit price equations:

p = 200 - 1/5(110) - 1/10(69)
= 200 - 22 - 6.9
= 171.1

q = 15 - 1/10(110) - 1/4(69)
= 15 - 11 - 17.25
= -13.25

Now, we can find the total revenue by multiplying the unit prices with their respective quantities:

R(110, 69) = (171.1 * 110) + (-13.25 * 69)
= 18,821 + (-913.25)
= $17,907.75

Therefore, R(110, 69) = $17,907.75.

2. R(61, 116):
Substituting the values of x = 61 and y = 116 into the unit price equations:

p = 200 - 1/5(61) - 1/10(116)
= 200 - 12.2 - 11.6
= 176.2

q = 15 - 1/10(61) - 1/4(116)
= 15 - 6.1 - 29
= -20.1

Now, we can find the total revenue:

R(61, 116) = (176.2 * 61) + (-20.1 * 116)
= 10,753.8 + (-2,333.6)
= $8,420.2

Therefore, R(61, 116) = $8,420.2.

Answer 3

To compute the values of R(x, y), we need to use the given unit prices and quantities demanded. The formula for total revenue is given by:

R(x, y) = p * x + q * y

where p is the unit price for the finished rolltop desks and q is the unit price for the unfinished rolltop desks.

Let's substitute the given unit prices and quantities demanded into the formula to find the values of R(x, y):

For R(110, 69):

p = 200 - (1/5) * 110 - (1/10) * 69
= 200 - 22 - 6.9
= 171.1

q = 15 - (1/10) * 110 - (1/4) * 69
= 15 - 11 - 17.25
= -13.25

Now, substitute these values into the formula:

R(110, 69) = (171.1 * 110) + (-13.25 * 69)
= 18,821 + (-913.25)
= 17,907.75

So, R(110, 69) is equal to 17,907.75 dollars.

For R(61, 116):

p = 200 - (1/5) * 61 - (1/10) * 116
= 200 - 12.2 - 11.6
= 176.2

q = 15 - (1/10) * 61 - (1/4) * 116
= 15 - 6.1 - 29
= -20.1

Now, substitute these values into the formula:

R(61, 116) = (176.2 * 61) + (-20.1 * 116)
= 10,742.2 + (-2,333.6)
= 8,408.6

So, R(61, 116) is equal to 8,408.6 dollars.