Question

A​ landscaper, who just completed a rectangular flower garden measuring 15 feet in length by 7 feet in​ width, orders 1 cubic yard of premixed​ cement, all of which is to be used to create a border of uniform width around the garden. if the border is to have a depth of 44 ​inches, how wide will the border​ be?

Answer 1

The width of the border will be 0.1648... ft or 1.9786... inches.

Step-by-step explanation

Lets assume, the width of the border is
`x` ft.

The rectangular garden is 15 ft in length and 7 ft in width. So, the area of the garden
`=(15*7)ft^2= 105 ft^2`

Now, the length of the garden including border
`=(15+2x) ft` and the width of the garden including border
`=(7+2x) ft`

So, the area of the garden including the border
`= (15+2x)(7+2x) ft^2`

Thus the area of the border
`=[(15+2x)(7+2x)-105] ft^2`

Total amount of premixed cement needed is 1 yard³ = 27 ft³

As the depth of the border is 44 inches or
`((44)/(12))ft` or
`((11)/(3))ft`, so....

`(11)/(3)[(15+2x)(7+2x)-105] = 27\\ \\ 11(105+44x+4x^2 -105)=81\\ \\ 11(4x^2+44x)=81\\ \\ 44x^2+484x-81=0`

Using quadratic formula....

`x= (-484+/-√(484^2-4(44)(-81)))/(2(44))\\ \\ x= (-484+/-√(234256+14256))/(88)\\ \\ x= (-484+/-√(248512) )/(88)\\ \\ x= 0.1648...`

(Negative value ignored)

So, the width of the border
`=0.1648...ft= (0.1648...*12)inches= 1.9786...inches`