confusing word problem thingg?

Question

a decorator plans to place a rug in a 9m by 12m room so that a uniform strip of flooring around the rug will remain uncovered. How wide will this strip be if the area of the rug is to be half the area of the room?

at first i was doing something like 9(12) then i got 108 but yea i dont know what im doing

Answer 1

Area of room = 9 X 12 = 108
Area of rug = 108 X 0.5 = 54

Let s be the width of the strip
Let t = 2s

Length of rug = 12 - t
Width of rug = 9 - t

area = length X width
54 = (12-t)(9-t) = 108 - 21t + t^2
t^2 -21t + 54 =
(t-18)(t-3) = 0
t = 18 or t = 3
s = 9 or s = 1.5

s cannot be 9 because the room is only 9X12

Therefore answer is 1.5m wide

Answer 2

For problems like this, always draw a picture. In this case, you'll need a picture of a rectangle within a rectangle. (The outside rectangle is the room; the inside rectangle is the rug.)

We know the length of the room is 12m, and the width of the room is 9m. Label the edges of the outside rectangle with these figures.

The thing your looking for is the width of the uncovered strip. Call this x. In your picture, x will be the distance between any edge of the inside rectangle and the corresponding edge of the outside rectangle.

What are the dimensions of the inside rectangle? The outside length is 12m, but there's a left boder that's x meters wide, and a right border that's x meters wide. So start with 12, and take away two x's. That's what's left to be the length of the rug: 12-2x. Similarly, the width of the rug is 9-2x.

You also know the area of the rug: since the room is 108 sq.m, and the rug is half of that, then the rug is 54 sq.m. Area = length times width. Thus:
(12-2x)(9-2x) = 54.

That's the setup. Solve this equation for x. (The equation gives two answers, but only one of them is valid for the picture you drew. Indicate the right one.)

Answer 3

So the rug has an area of 54 m². Let the strip around the rug be x meters wide. Then the rug will be (9-2x) by (12-2x) and this gives 108 - 42x + 4x² = 54 or (in standard form) 4x² - 42x + 54 = 0 Factoring out the 2 gives
2x² - 21x + 27 = 0 Solving the quadratic gives the width of the strip as 1.5 meters.


Doug