SOLUTION: A rectangular park measuring 31 yards by 27yards is surrounded by a trial of uniform width. If the area of the park and the trial combined is 2205 square yards, what is the width o

Question 964092: A rectangular park measuring 31 yards by 27yards is surrounded by a trial of uniform width. If the area of the park and the trial combined is 2205 square yards, what is the width of the path?
Found 2 solutions by addingup, macston:
Answer by addingup(3677) (Show Source): You can put this solution on YOUR website!
1) The trail around the park is all the same width, like the frame on a picture.
2) The area of a rectangle = l x w
3) A rectangle has 2 lengths and 2 widths. Let's call the width of the trail x:
(31+2x)(27+2x)= 2205 Multiply (FOIL)
31*27+31*2x+2x*27+4x^2 = 2205
= 837+62x+54x+4x^2 = 2205
=837+116x+4x^2 = 2205 Let's rewrite as a quadratic polynomial:
4x^2+116x+837= 2205 Divide both sides by 4:
Now solve by completing the square. Subtract 837/4 from both sides:

Now divide on right:
Subtract 342 from both sides:
Factor the equation:
Now we have two possible answers
or
or We know our answer cannot be negative, so we have that the trail is 9 wide. Let's prove it
Length: 31+ 2(9)= 49
Width: 27+2(9)= 45
Area of the park with the trail:
49*45= 2205 We've got the correct answer. See the sketch I made you:

Answer by macston(5194) (Show Source): You can put this solution on YOUR website!
x=width of path

ANSWER: The width of the path is 9 yards.
CHECK:

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