area and perimeter

Question

A rectangle is 3 times as long as it is wide and has the same perimeter as a square whose area is 4 square feet larger than that of the rectangle. What are the dimensions of both the rectangle and the square?

Answer 1

Let L = length of rectangle

Let W = width of rectangle

Let S = side of square

 

The rectangle is 3 times as long as it is wide  ==>  L = 3W  

 

The rectangle has the same perimeter as the square  ==>  2L + 2W = 4S  

 

The area of the square is 4 square feet larger than that of the rectangle  ==>  S^2 = LW + 4  

 

Plug the value for L from the 1st equation in to the 2nd equation to get

2(3W) + 2W = 4S

6W + 2W = 4S

8W = 4S

2W = S

 

Plug this value for S (S = 2W) and the value for L (L = 3W) in to the 3rd equation to get

 

(2W)^2 = (3W)W + 4

4W^2 = 3W^2 + 4

W^2 = 4

W = +- 2

 

The width cannot be negative so discard the W = -2 answer

 

So W = 2, L = 6 and S = 4

 

The rectangle is 2 feet by 6 feet and the square is 4 feet by 4 feet.