Question

Matthew has 170 tiles he can use for this project. Identify the largest patio design that he can make. Show or explain your reasoning.

Answer 1

• Design 12

Explanation:

• See the attached for missing part of the question

As we see each design comprises of entrance and exit - each one tile and:

• Design 1 has 1 row and 3 columns,
• Design 2 has 2 rows and 4 columns,
• Design 3 has 3 rows and 5 columns.

So design x has x rows and two more columns according the pattern we observe.

The number of tiles of design x is:

• x by (x + 2) and 2 more tiles

We can put it as equation to get 170 tiles and solve for x:

• x(x+2) + 2 = 170
• x² + 2x - 168 = 0
• x² - 12x + 14x - 168 = 0
• x(x - 12) + 14(x - 12) = 0
• (x + 14)(x - 12) = 0
• x = - 14 (discarded as negative root)
• x = 12 (the solution)

So this is a design 12

It has 12 rows, 14 columns, 1 entrance and 1 exit tiles, the total number is

• 12*14 + 2 = 170 tiles