SOLUTION: A decorator plans to place a rug in a room 9 m by 12 m in such a way that a uniform strip of flooring around the rug will remain uncovered. If the rug is to cover half of the floor

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Question 171739This question is from textbook Algebra and Trigonometry: Structure and Method-Book 2
: A decorator plans to place a rug in a room 9 m by 12 m in such a way that a uniform strip of flooring around the rug will remain uncovered. If the rug is to cover half of the floor space, what should the dimensions of the rug be? This question is from textbook Algebra and Trigonometry: Structure and Method-Book 2

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Call the width of the uniform space around the
rug x
The dimensions of the rug will then be
%289+-+2x%29%2A%2812+-+2x%29
Half of the area of the 9 x 12 room is
%289%2A12%29%2F2+=+54
54+=+%289+-+2x%29%2A%2812+-+2x%29
54+=+108+-+24x+-+18x+%2B+4x%5E2
54+=+108+-+42x+%2B+4x%5E2
4x%5E2+-+42x+%2B+54+=+0
Solve using quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a+=+4
b+=+-42
c+=+54
x+=+%28-%28-42%29+%2B-+sqrt%28+%28-42%29%5E2-4%2A4%2A54+%29%29%2F%282%2A4%29+
x+=+%2842+%2B-+sqrt%28+1764+-+864+%29%29+%2F+8+
x+=+%2842+%2B-+sqrt%28+900+%29%29+%2F+8+
x+=+%2842+%2B+30%29+%2F+8+
x+=+%2842+-+30%29+%2F+8+
------------------------
The answers are:
x+=+9
x+=+1.5
The 1st answer is impossible, so the width of
the border around the rug is 1.5 m
The dimensions of the rug are
%289+-+2%2A1.5%29%2A%2812+-+2%2A1.5%29
%289+-+3%29%2A%2812+-+3%29
6%2A9+=+54
The dimensions of the rug are 6m x 9m