Question

A pool measuring 8 meters by 26 meters is surrounded by a path of uniform​ width, as shown in the figure. If the area of the pool and the path combined is 1008 square​ meters, what is the width of the​ path?

Answer 1

Let the width path be x.
Length of the outer rectangle = 26 + 2x.
Width of the outer rectangle = 8 +2x.

Combined Area = (2x + 26)*(2x + 8) = 1008

2x*(2x + 8) + 26*(2x + 8 ) = 1008

4x² + 16x + 52x + 208 = 1008

4x² + 68x + 208 - 1008 = 0
4x² + 68x - 800 = 0.          Divide through by 4.
x²  + 17x - 200 = 0 . This is a quadratic equation.

Multiply first and last coefficients:  1*-200 = -200

We look for two numbers that multiply to give -200, and add to give +17

Those two numbers are 25 and -8.

Check:   25*-8 = -200         25 + -8 = 17

We replace the middle term of +17x in the quadratic expression with 25x -8x


 x² +17x - 200 = 0     

x² + 25x - 8x - 200 = 0     

x(x + 25) - 8(x + 25) = 0

(x+25)(x -8) = 0

x + 25 = 0    or   x - 8 = 0

x = 0 -25              x = 0 + 8

x = -25                    x = 8

The width of the path can not be negative.

The only valid solution is x = 8.

The width of the path is 8 meters.