Finding area and dimensions

Question

A boy wants to build rectangular cages. He wants all the sides on the base of his cages to beat least 10 cm long.he also wants the base area to equal 576 cm2. The boy needs help to find all the possible lengths and widths for the base of the cages. He used only whole centimeters, with no fractional parts.what are all the possible dimensions for the rectangular base?

Answer 1

To find possible combinations, break 576 up into a prime number tree

 

                                        576

                                       /       \

                                    24        24

                                   /   \       /   \

                                  4    6     4    6

                                 / \  / \    / \  / \

                               2  2 2  3  2 2 2 3

 

Then, to find possible side lengths, multiply different combinations of these prime numbers together (make sure each pair adds up to 10)

 

1)  3x3x2= 18cm            2x2x2x2x2= 32cm

 

2)  3x3x2x2= 36cm         2x2x2x2= 16cm

 

3)  3x2x2= 12cm            3x2x2x2x2= 48cm

 

4)  3x2x2x2= 24cm         3x2x2x2= 24cm

 

Answer 2

Let's factor out 576:

576 = 2 * 288

576 = 2 * 2 * 144

576 = 2 * 2 * 2 * 72

576 = 2 * 2 * 2 * 2 * 36

576 = 2 * 2 * 2 * 2 * 2 * 18

576 = 2 * 2 * 2 * 2 * 2 * 2 * 9

576 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3

576 = 2⁶ * 3²

 

We can create some baselines with this information on the minimal lengths and widths:

(2 * 2 * 2 * 2) * (3 * 3 * 2) with one factored 2 remaining that can be in either

(2 * 2 * 2 * 3) * (3 * 2 * 2) with one factored 2 remaining that can be in either

 

That gives us the following possibilities:

(2⁵) * (3² * 2) and (2⁴) * (3² * 2²)

(2⁴ * 3) * (3 * 2²) and (2³ * 3) * (3 * 2³)

 

That leaves the dimensions as (respectively):

32 cm by 18 cm, 16 cm by 36 cm,

48 cm by 12 cm, 24 cm by 24 cm