Question

Help please! I dont just want the answer. I need someone to help me through it!

A square picture frame occupies an area of 112 ft2. What is the length of each side of the picture in simplified radical form?

Answer 1

The area is 112.
Let the length of a side be x. So what is our equation that we have about the picture frame?

Answer 2

Im really not sure @Nory

Answer 3

would 112 ft. 2 be 112^2? @Nory

Answer 4

The area of the frame is 112, but it is also x*x (because of the area of the rectangle.) So our equation would be x^2 = 112.
Do you know how to solve this equation?

Answer 5

wouldnt that equation turn into x^2 - 112? @Nory

Answer 6

Yes, but there's an easier way to solve it.
Go back to x^2 = 112 and take square roots of both sides. What do you get?

Answer 7

square root of 112 is 10.583? and x^2 is 2.718

Answer 8

@Nory

Answer 9

Write it as a radical. x = √122
x is the length of a side.

Answer 10

Oh wait, 112. Not 122. What am I thinking?

Answer 11

haha its okay. so x = the square root of 112? which is 10.583?

Answer 12

@Nory

Answer 13

@jim_thompson5910 could you help me?

Answer 14

can you factor 112 where one factor is a perfect square

Answer 15

yes?

Answer 16

how so

Answer 17

you have to find a number that can be multiplied with another to get 112

Answer 18

yep

Answer 19

ex: 112 = 2*56

but neither factor is a perfect square

Answer 20

i dont get how you find out if a number is a perfect square.

Answer 21

it might help to list out the perfect squares (up to 112)

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Answer 22

so i would find the best perfect square you can divide out of 112?

Answer 23

so for instance, 9 is NOT a factor since 112/9 = 12.4444

Answer 24

it would be a factor if you got a whole number when you divided

Answer 25

so the number it equals would have to be a whole number?

Answer 26

yes if you divided and got a whole number, then the number you divided by is a factor

Answer 27

ex: 112/2 = 56, so this means 2 is a factor of 112

it also means 56 is a factor as well

Answer 28

but neither are perfect squares

Answer 29

im so confused

Answer 30

do you see how 2 is a factor of 112?

Answer 31

so 56 and 2 are not perfect squares because their square roots are not whole numbers?

Answer 32

exactly

Answer 33

25 is a perfect square because the square root of 25 is 5 (a whole number)

Answer 34

so i have to find a number that is a factor of 112 and is also a perfect square?

Answer 35

exactly

Answer 36

and you want to find the largest possible factor that's a perfect square

Answer 37

okay so the largest number is 16?

Answer 38

yep good

Answer 39

so 112/16 = ???

Answer 40

7

Answer 41

this means that 112 = 16*7

Answer 42

so each side is 7?

Answer 43

and this further means....


\[\large \sqrt{112} = \sqrt{16*7}\]


\[\large \sqrt{112} = \sqrt{16}*\sqrt{7}\]


\[\large \sqrt{112} = 4\sqrt{7}\]

Answer 44

notice how I simplified by exploiting the fact that you can factor where one factor is a perfect square (the largest perfect square possible)

Answer 45

yea you got 4. so i have to simplify 16 to finish the equation?

Answer 46

yes you have to take the square root of 16 to get 4

Answer 47

this was easier then I thought! thank you once again!

Answer 48

yw