Question

solve & discuss it!
matt and his dad are building
a tree house. they buy enough
flooring material to cover an area
of 36 square feet. what are
all possible dimensions of
the floor?
look for relationships
can different floor dimensions
result in the same area?

Explanation:

Step1: Define area formula

For a rectangular floor, area $A = l \times w$, where $l$ = length, $w$ = width, and $A=36$ sq ft.

Step2: List positive factor pairs

Find all pairs of positive integers whose product is 36:
$1 \times 36 = 36$
$2 \times 18 = 36$
$3 \times 12 = 36$
$4 \times 9 = 36$
$6 \times 6 = 36$
We also consider non-integer positive real numbers: for any positive real number $l$, $w = \frac{36}{l}$.

Step3: Address relationship question

Different dimension pairs (e.g., 4×9 and 6×6) multiply to the same area, so yes, different dimensions can result in the same area.

Answer:

All positive real number pairs $(l, \frac{36}{l})$ where $l$ is the length (in feet) and $\frac{36}{l}$ is the width (in feet). The integer dimension pairs are:

1. 1 foot by 36 feet
2. 2 feet by 18 feet
3. 3 feet by 12 feet
4. 4 feet by 9 feet
5. 6 feet by 6 feet

Yes, different floor dimensions can result in the same area.