# Math Help - Word problem

1. ## Word problem

Q) A rectangular field , 70m long and 50m wide, has a path of uniform width around it .If the area of the path is 1024m^2 , find the width of the path.

Solution let x be the uniform width of the path

According to the given condition,

(50+2x)(70+2x)-1024=50*70

...............................

and the answer comes out to be 4m after solving this.

I have problem in that how the bold equation is formed ?

2. You know the area of a rectangle to be $A = lw$ and so $A_{f} = 50 \times 70 \text{ m}^2$

Like they did, let x be the width of the path.

Now consider the length of the field and path - it is going to be the length of the field (70) plus the width of the path at both ends (2x) which is equal to $l = 70+2x$
Do the same for the width and the path: $w = 50+2x$

The area of the field+path is also length x width: $A_{f+p} = (50+2x)(70+2x)$

The area of the path will be the area of the field+path minus the area of the filed: $A_p = A_{p+f} - A_f$ - you are given that $A_p = 1024$ and so the equation is $1024 = (50+2x)(70+2x) - 50 \times 70$

Which can be rearranged into $(50+2x)(70+2x) - 1024 = 50 \times 70$