Dimensions- Square Frames

[Guest]

A length of wire is used to construct two square wire frames. The area enclosed by one frame is one-half the area enclosed by the other. Calculate dimensions of the two frames.

Length of wire - 100 inches.

Disregard thickness of wire.




Area of square - S^2

Let X by X be area of smaller square and Y by Y be area of larger square.

X^2 = Y^2 / 2

How do I proceed from this point ?

[Guest]

Don't have any more unknowns declared than you need....
There is only one thing you have to find out.....the length of one side of any of the squares and the size of the other can be found from it.
So if we let Y = length of the side of the small square. The length of the large square side will be (SqRoot of 2 times Y). (SqRoot of 2 is 1.414)
This is because we are told the area of one square is twice the area of the other...and notice we have only 1 unknown....Y...so only equations with 1 unknown needed..............

[Guest]

Y = side of smaller square

1.414Y = side of larger square

???

[Guest]

Yes, that is correct. These statements describe the lengths of the sides of each square, described in terms of Y. Now make use of another fact given in the question as well as the information in these 2 statements to set up an equation describing the fact given. Then this equation can then be solved to find an actual value for Y

[Guest]

Perimeter of squares -

2L + 2W = P

2L + 2W = 100

L + W = 50

50 - L = W

I don't think that is correct.

[Guest]

If you do not think it is correct, why did you write it?.
Did you read the last post?......and if you did how are you making use of the information from the 2 statements in an earlier post and another fact given in the question.

The suggested way to proceed was in the sentence copied below.....
"Now make use of another fact given in the question as well as the information in these 2 statements......"

All you have done is give equations that relate to any rectangle.......and are now using different symbols....
The shapes in the question are squares.........you didnt even try to resolve that based on the sizes you had worked out before..................

[Guest]

Sorry for last post.

Perimeter for square-

P = 4S

4Y (perimeter for smaller square)

4(1.414Y) (perimeter for larger square)

4Y + 4(1.414Y) = 100

4Y + 5.656Y = 100

9.656Y= 100

9.656Y/ 9.656 = 100/9.656

Y = 10.356255 = 10.36

10.36 * 10. 36 = 107.3296 = 107.33 in.^2 (smaller square)

107.33 * 2 = 214.66 in.^2 (larger square)

I am guessing. If that is not correct I don't know.

[Guest]

Yes this bit is correct........
=========
Perimeter for square-

P = 4S.....this is just general statement...perimeter is 4 times the sides...OK

4Y (perimeter for smaller square)

4(1.414Y) (perimeter for larger square)

4Y + 4(1.414Y) = 100......yes this is the equation that describes the length of wire used to make both squares in terms of the size of the smaller square "Y" and you are told it was 100 so you let the equation equal 100 and solved it....OK

4Y + 5.656Y = 100

9.656Y= 100

9.656Y/ 9.656 = 100/9.656

Y = 10.356255 = 10.36.....this is the length of side of smaller square......OK
=============

You were asked in the question to find the dimensions of the squares....what you did after this was work out areas....
Dimensions means the lengths of the sides........
After finding Y above for the smaller square, you could have calculated the length of side of larger square directly...
So go ahead and do it now......
Then you can work out the areas of each square if you want and see is one twice the area of the other as a check that you are correct..............
.......Have you figured out yet why the sides of the larger square is 1.414 times the sides of the smaller square...????.......

Regards your answer in general, you should work logically through it and have some idea of your methods, rather than saying "I am guessing" and "I don't know"......etc.......so that you can apply what you have learned to the next question......

[Guest]

(1.414Y)

(1.414 * 10.35625) = 14.64373 = 14.65

10.36 * 10.36 = 107.3296 = 107.33
14.65 * 14.65 = 214.6225 = 214.63

214.63 / 2 = 107.31

"Have you figured out yet why the sides of the larger square is 1.414 times the sides of the smaller square...????......."

Twice the area ?

[Guest]

Yes, you have got the sums done now......and you were able to first find the length of side of the small square and then work out the length of side of the large square.......without needing to work out the areas.....
That was the reason for me asking this question in the last post .........
"Have you figured out yet why the sides of the larger square is 1.414 times the sides of the smaller square...????......."

How did I know at the start that before working anything out that one squares sides would be 1.414 times longer than the length of the others sides.....?????......

[Guest]

I have not figured it out.

[Guest]

Well then, why did you use 1.414 in these expressions copied below.........
Where did the 1.414 come from and how does it relate to this type of problem.........
What is common about all problems of this type.................

=======
Y = side of smaller square

1.414Y = side of larger square......how do you know that...???
===========
4Y (perimeter for smaller square)

4(1.414Y) (perimeter for larger square)

4Y + 4(1.414Y) = 100

============================

[Guest]

Your words:

"So if we let Y = length of the side of the small square. The length of the large square side will be (SqRoot of 2 times Y). (SqRoot of 2 is 1.414)"


I added the sides together to obtain 100, due to 100 inches of wire was used to construct two frames.

"Where did the 1.414 come from and how does it relate to this type of problem.........
What is common about all problems of this type................."

I don't know why the square root of 2 is used for the larger.

Because information about more than one item is being determined ?

[Guest]

Well from reading the question........
A length of wire is used to construct two square wire frames. The area enclosed by one frame is one-half the area enclosed by the other. Calculate dimensions of the two frames.

Length of wire - 100 inches.

...........You should see from the question that one frame is one-half the area of the other......
That means that one frame must be twice the area of the other.......
The smaller is half of the larger OR the larger is twice the smaller....whichever way you want to say it.....

For example......
If the smaller had 1 inch sides and the larger had 2 inch sides
Their dimensions would be 1 inch by 1 inch for the smaller and 2 inches by 2 inches for the larger....
Their areas would be 1 square inch for the smaller and 4 square inches for the larger.....
You could say the smaller is 1/4 the area of the larger OR the larger is 4 time the area of the smaller......

If the smaller was 2 by 2 and the larger was 4 by 4.....Ratio 2 : 1 for sides length...
The smaller area would 4 sq inches and the larger would be 16 sq inches
So as before the smaller id 1/4 the area of the larger OR the larger is 4 times the smaller. Ratio 4 : 1 for areas

Also in reverse....If we know the area we can calculate the length of the side......
For the smaller 4 sq inches area....the length of the sides will be (sq root of 4) = 2 inches
and for the larger 16 sq inches area....the length of the sides will be (sq root of 16) = 4 inches.
and we can see the ratio of the lengths of the side is 2 : 4 or the smaller is half the length of the largers side OR the Larger frame has sides twice the size of the smaller.
So if the areas are in the ratio of 4 : 1 then the sides will be in ratio of (sq root of 4 : 1) or 2 : 1.

But in the question the ratio of the areas of the frames is 2 : 1 or 1 : 2 so the ratio of the sides length will be (sqroot 2 : 1)
So that gives 1.414 :1 so if Y is the side of the smaller then the larger will be 1.414Y inches

[Guest]

I don't understand the ratios.

[Guest]

Ratio is just how we compare the the size of 2 things........for example
If you buy a 1 gallon jar of milk and I buy a 2 gallon jar of milk....

The ratio of the size of the jars is 1 to 2 but it is normally written as 1 : 2

My jar is twice the size of yours...or.. yours is half the size of mine.......

You don't really need this to solve the squares problem.....

For any square if you know the area of it then the length of its sides will be the square root of its area

The question told us one square was half the area of the other or said the other way around one square is twice the area of the other.......
We let the length of the side of the smaller one equal "Y"
We actually jumped a step in the question because the figures were so simple....
If we had let the small square have an area of Y^2 (Y squared) then the other larger area would be 2Y^2 (two Y squared)..
and because we were told the length of the total perimeter was 100 we needed to set up and equation in linear dimensions so we needed to know the length of the sides of each square in terms of "y".....
We know the length of the sides of a square is the square root of its area.....

So square root of Y^2 is simply "Y".....and.......Square root of 2Y^2 is simply....Square root of 2 time "Y".....sqrt2 x Y
and this is 1.414 Y what we got before..............

This is where square numbers come from.....eg....5 x 5 = 25.......sqrt25 = 5...............

[Guest]

I understand now. Thanks.

[Guest]

Generally problems like this are solved by algebra......let Y = something unknown and make equations to solve it.
But it could have been done numerically using the ratios we mentioned ..... if you wanted.

We know the areas of the frames are in ratio of 2 : 1
So sides will be in ratio of sqrt2 : sqrt 1 that is same as saying .... 1.414 : 1
All the sides of each add up to perimeter of each so the perimeters of each will also be in the ratio of 1.414 : 1 as well.

If we consider this as parts of the total wire used..... 1 part for the small frame and 1.414 parts for the large frame.....
That means 2.414 parts all together......

That means the fraction of wire used for the small frame will be 1/2.414 of the 100 inches
and the fraction used for the large frame will be 1.414/2.414 of the 100 inches

So the perimeter of the small frame is 100/2.414 = 41.425 inches so each side is 41.425/4 = 10.356 inches as before
And the perimeter of the large frame is 100(1.414/2.414) = 58.575 inches so each side is 58.575/4 = 14.644 inches as before

And no algebrs in sight...............but maybe not any easier...?.?.?.....

[Guest]

Thanks for additional information with ratios. Maybe not any easier.