Can you solve this algebra problem? It was a test bonus question?

Question

A rectangular swimming pool measures 12m long X 8m wide. A uniform border is to be built around it using 120 sq. meters of tile, all of which have to be used. How wide should the border be? If at least a 2-meter border is required, can it be done with the available tile?

Answer 1

There is a more direct way of solving this problem. The pool's perimeter is 2 (w + l) m. So the length of tile needed to go around the edges of the pool is 2 (12 + 8) m = 2 (20) m = 40 m. If the border is to be at least 2 m wide, then the entire area of tile needed along the edges of the pool is 2 m x 40m = 80 m².

The previous calculations only take into consideration the tile along the perimeter of the pool. We still have to consider the area of the tile to be placed at the corners of the pool. If the border is 2 m wide, then each corner piece is (2 m)² = 4 m² in dimension. Since there are 4 corners, we multiply 4 m² by 4 to obtain 16 m² needed to cover the corners. Adding all this up, we get (80 + 16) m² = 96 m², which means 120 m² is more than enough to do the job.

Mathematically, we can express the area of tile needed for the border this way:

A = [2 (w + l)] w'+ 4 (w')²,

where w is the width of the pool, l is the length of the pool and w' is the width of the tile to be laid around the pool.

Answer 2

The border's width would be x, making the border's dimensions (including the pool) equal to (12 + 2x)(8 + 2x) or by FOIL,

96 + 40x + 4x^2 which equals the pool's area of 96 + the border's area of 120 = 216 sq m.

Subtract the 216 from both sides to get -120 + 40x + 4x^2 = 0

Divide by 4 to get -30 + 10x + x^2 = 0

Solve for x using the quadratic formula and see if it's more or less than 2. Also check for typos- it's very early.