A rectangular garden is 10 feet by 4 feet. A gravel border with uniform width along the sides and right angle corners surrounds the garden.
The area of the gravel border is six times the area of the garden. What is the perimeter of the outside of the gravel border?
Let the width of the gravel border be x.
Then the rectangle's sides after the gravel is:
Length = 10 + x
Width = 4 + x
The area of the rectangle + the border = (x+10)*(x+4)
==> The area of the border = (x+10)(x+4) - the area of the rectangle
==> The area of the border = (x+10)(x+4) - 10*4
==> The area of the border = x^2 + 14x + 40 - 40 = x^2 + 14x
==> But we are given that the area of the gravel border = 6 times the area of the garden
==> x^2 + 14x = 6*40
==> x^2 + 14x = 240
==> x^2 + 14x -240 = 0
Now we will factor and solve for x.
==> (x- 10)(x+24) = 0
==> x = 10
==> x = -24 ( Not valid )
Then the width of the border is x = 10
Then the length of the garden now = 10+10 = 20
The width of the garden with the border = 14
Then the perimeter = 2*20 + 2*14 = 40 + 28 = 68
Then, the perimeter of the gravel border = 68 ft