# 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) = ?

## 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

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## 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.

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## 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.