Why is the area 76m^2?

Question

George built a swimming pool in his backyard. The dimensions of the pool are: Length: 24 - 2xm & Width: 16 - 2xm 

 

The tiled path around the outside of the pool is x meters wide. Using the binomial expansion we need to find: 

a.) the area of the swimming pool

b.) the area of the path

 

This expression is the expanded form for the area of the pool: (24 - 2x)(16 - 2x) = 384 - 80x + 4x² 

 

The area of the path is whatever is left when we subtract the pool from the total:

Area of path : 384 - (384 - 80x + 4x²)m² = 80x - 4x²m²

 

George measures the width of the path & discovers its 1m. What's the area of the pool & path now?

Substituting x = 1 into: 384 - 80x + 4x² gives: 384 - 80 + 4 = 308m² therefore the area of the path = 76m².

 

Why is the area 76m²?

Answer 1

When you were finding the area of walkway, you took the area around walkway border.  If you draw the pool and walkway, you should have 2 rectangles.  A small rectangle to represent the pool, and a larger rectangle surrounding the smaller rectangle.  This large rectangle represents the walkway border.  Not the walkway itself.  The area between the walkway border and pool is actually the walkway.

 

So the area of walkway = area of walkway border - area of pool

 

If the length of the walkway border is 24 m and the width of the walkway border is 16 m, then area of walkway border is

 

24 m * 16 m = 384 m²

 

 

Area of walkway = 384 m² - 308 m² = 76 m²