Find the Width

Question

Julie Stone designed a rectangular patio that is 25ft by 40ft. This patio is surrounded by a terraced strip of uniform width planted with small trees and shrubs. If the area A of this terraced strip is 504 square ft, find the width x of the strip. 

Answer 1

(2X + 40)(2X + 25) = 504 + 1000   (40 x 25 = 1000 area of rectangular patio)

 

 4X^2 + 80X + 50X = 1504     Subtract 1504 from both sides.  

 

4X^2 + 80X + 50X - 1504       4X^2 + 130X - 1504 = 0     2(2X^2 +65X - 752) = 0 

 

  

3.5 for X would work. 

  

 

 

Answer 2

I believe the answer will have to depend on whether or not the patio is attached to a house on one or more sides and on which side(s). 

 

But in short, the terrace is a strip that surrounds the patio, and you can break it up into smaller rectangles. It's easier to visualize if you draw it out but imagine just the pieces of the terrace that are in essence "extensions" of the patio. Those would have dimensions of width x dimension of the patio it shares a side with. That would leave you the corner pieces of the terrace that just touch the corners of the patio. Those would share sides with the pieces of the terrace you considered above, specifically the sides that do not touch the patio or in other words, the width. So the corner pieces are squares with sides equal to the width of the terrace. 

 

So if the terrace surrounds the patio completely, you have 4 "extension pieces" of the terrace and 4 "corner pieces" in total.

 

That means you have 2 extension pieces of dimension:

 

width x 25ft

 

and 2 extension pieces of dimension:

 

width x 40ft

 

and 4 corner pieces of dimension:

 

width x width

 

adding those up you have:

 

2 x (width x 25) + 2 x (width x 40) + 4 x width x width = 504 sqft

 

solving for width (w):

 

4w^2+130w=504 <--- this is a quadratic so you can either try to factor and solve for a zero or using quadratic equation:

 

2w^2+65w-252=0

 

w = -36 or 3.5

 

Since you can't have a negative width, 3.5ft is the answer.