Abey Numkena is an interior designer. She has been asked to locate an oriental rug for a new corporate office. As a rule, the rug should cover 1/2 of the total floor area with a uniform width surrounding the rug.

a. if the dimensions of the room are 12 feet by 16 feet, write an equation to model the situation.

b. what are the dimesions of the rug?

Let L be the long dimension of the rug and W be the narrow dimension.

L W = (1/2) 12 x 16 = 96
For equal borders all around,
16 - L = 12 - W
Solve those two simultaneous equations.
L - W = 4 = L - 96/L
L^2 -4L +96 = 0
(L-12)(L+8) = 0
Choose the positive root.

a. To model the situation, let's assume that the width of the uniform area surrounding the rug is represented by 'x'. The dimensions of the rug will be '12 - 2x' (subtracting twice the width from the original length) and '16 - 2x' (subtracting twice the width from the original width). The equation to model the situation is:

(12 - 2x) * (16 - 2x) = (1/2) * 12 * 16

b. To find the dimensions of the rug, we can solve the equation from part a.

(12 - 2x) * (16 - 2x) = (1/2) * 12 * 16

Expanding the equation:

192 - 40x + 4x^2 = 96

Rearranging the equation and setting it equal to zero:

4x^2 - 40x + 96 = 0

Using factoring or quadratic formula, we can solve for 'x'.

a. To model the situation, we can use the equation: (Length of the rug) x (Width of the rug) = (1/2) x (Total floor area)

Let's calculate the total floor area of the room. In this case, the dimensions of the room are given as 12 feet by 16 feet. Therefore, the total floor area can be found by multiplying the length and width of the room.

Total floor area = (Length of the room) x (Width of the room)
Total floor area = 12 ft x 16 ft
Total floor area = 192 square feet

Now, let's write the equation for the rug dimensions:
(Rug length) x (Rug width) = (1/2) x (Total floor area)

Let's represent the rug length and rug width as 'L' and 'W' respectively.

L x W = (1/2) x 192

b. To find the dimensions of the rug, we need to solve the equation from part a for the unknowns L and W.

L x W = (1/2) x 192

Since we do not have any other information about the specific dimensions or constraints, we can have multiple possibilities.

Here are a few possible solutions:

Option 1:
Assuming the width of the rug is W and it is uniform surrounding the rug, we can say the length of the rug is 16 feet since the width of the room is 16 feet.

L x W = (1/2) x 192
16 x W = (1/2) x 192
16W = 96
W = 6

So, the dimensions of the rug are 16 feet by 6 feet.

Option 2:
Assuming the length of the rug is L and it is uniform surrounding the rug, we can say the width of the rug is 12 feet since the length of the room is 12 feet.

L x W = (1/2) x 192
L x 12 = 96
L = 8

So, the dimensions of the rug are 8 feet by 12 feet.

Please note that there can be other possible solutions depending on the specific requirements or constraints for the rug.