Rectangle - Proportion - Solution/Explanation

[Guest]

A rectangle is most pleasing when the length and width are in the following proportion:

W / L = L / L + W

According to the proportion, if the perimeter is 8, then L + W = 4 *

If length is represented by X , then width is 4 - X *

Substituting in proportion 4 - X / X = X / 4 *

Multiplying by 4X = 16 - 4X = X^2

* I don't understand these steps.

[Guest]

If perimeter is 8 then 2(L + W) = 8
giving.... L + W = 4
and gives..... W = 4 - L ...hence... if length is X then W = 4 - X

So substituting in.... (4 - X) / X = X / X + (4 - X)
which gives ... (4 - X) / X = X / 4

to get rid of fractions multiply by across by 4X
gives.... 4X(4 - X) / X = 4X[X / 4]
gives .....4(4 - X) = 4X^2 / 4
an finally....cancel the 4 on RHS gives 16 - 4X = X^2

OR instead you could cross multiply... this.. (4 - X) / X = X / 4

giving ....4(4 - X) = X^2
then.... 16 - 4X = X^2

[Guest]

"So substituting in.... (4 - X) / X = X / X + (4 - X)
which gives ... (4 - X) / X = X / 4"

(4 - X) / X = X / 4 - I don't understand this step.

[Guest]

W / L = L / L + W .....original

then we worked out that if length which is L is actually X then W = 4 - X

so substitute into the original equation

(4 - X) / X = X / X + (4 - X) ...this is the same equation as original except W = 4 - X and L = X
So the equation is now in terms of X only

[Guest]

(4 - X) / X = X / X + (4 - X) ...this is the same equation as original except W = 4 - X and L = X

(4 - X) / X = X / X + (4 - X)....its probably this bit that is the problem

Written with more brackets.....(4 - X) / X = X / [ X + (4 - X)]

gives......(4 - X) / X = X / [ 4 ] ....the +X and -X cancel

[Guest]

I understand it now. Thanks.

[Guest]

Interested in solving the quadratic:

16 - 4X = X^2

X^2 + 4X - 16 = 0

A = 1
B = 4
C = -16

X = -B + - sq. rt. B^2 - 4(A)(C) / 2A

X = -4 + - sq. rt. 4^2 - 4(1)(-16) / 2

X = -4 + - sq. rt. 16 + 64 / 2

X = -4 + - sq. rt. 80 / 2

X = -4 + - 8.944 = 9 / 2

X = -4 + - 9 / 2

X = -4 + 9 = 13 / 2 = 6.5

X= -4 - 9 = -13 / 2 = -6.5

I don't believe that is correct.

[Guest]

X = -4 + - sq. rt. 16 + 64 / 2
it should be all over 2

X =[ -4 + - sq. rt. 16 + 64 ] / 2

X = 2.47 = L
so W = 1.53
perimeter = 8

[Guest]

Thanks.

Did you use the perimeter formula or proportion to find width ?

[Guest]

W / L = L / L + W .....original states that the ratio of W to L is same as L to (L + W)
OR the ratio of width to length is same as length to half the perimeter

then we worked out that if length which is L is actually X then W = 4 - X
which is an equation in terms of W and X

From the quadratic we solved for X which was same as the length
Then used the equation involving W and X to find W
W = 4 - 2.47
W = 1.53

and L + W = 2.47 + 1.53 = 4
so perimeter is 8 as was given for this instance of the formula

[Guest]

then we worked out that if length which is L is actually X then W = 4 - X
which is an equation in terms of W and X

I should have said....
then we worked out that if length which is L is actually X and if the perimeter is 8 then W = 4 - X
which is an equation in terms of W and X

[Guest]

I understand now.

A question:

What is the reasoning behind the statement that a rectangle is most pleasing when used in that proportion ?

[Guest]

It must be the golden ratio thing.....

Quote from Wikipedia etc.......

https://en.wikipedia.org/wiki/Golden_ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. If a line segment of length L and another line segment of length W is joined to the end of L. The total length is L + W.
Then L + W is to L as L is to W. .... OR its reciprocal W is to L as L is to L + W. ....as in the question.


If the ratio of L to W is the golden ratio
A golden rectangle with longer side L and shorter side W, when placed adjacent to a square with sides of length W, will produce a similar golden rectangle with longer side L + W and shorter side W. This illustrates the relationship W / L = L / (L + W) in the form of a rectangle.

[Guest]

Thanks again.