figuring out height/width from area

hello everyone

i have a rectangle that is 14*18

which is 252 sq. in.

i need to get the H&W of a rectangle that its area is some percent of 252 (lets say half)

so what will the H&W be for 126 sq.in. if it would have the same ratio as 14*18?

thank you very much

i need the formula, not the answer for this specific example

Re: figuring out height/width from area

Quote:

Originally Posted by

**yisman** hello everyone

i have a rectangle that is 14*18

which is 252 sq. in.

i need to get the H&W of a rectangle that its area is some percent of 252 (lets say half)

so what will the H&W be for 126 sq.in. if it would have the same ratio as 14*18?

thank you very much

i need the formula, not the answer for this specific example

1. Let H, W denote the original height and width of the rectangle and h, w the height and width of the transformed rectangle. Then you know

$\displaystyle \dfrac hH = k~\implies~h = k \cdot H$

$\displaystyle \dfrac wW = k~\implies~w = k \cdot W$

2. The area of the original rectangle is: $\displaystyle A_o = H \cdot W$

and the area of the transformed rectangle is:

$\displaystyle A_t = h \cdot w = (k \cdot H) \cdot (k \cdot W) = k^2 \cdot A_o~\implies~k=\sqrt{\dfrac{A_t}{A_o}}$

3. With your example:

$\displaystyle A_o = 252$

$\displaystyle A_t = 126$

Plug in into the equation above:

$\displaystyle 126 = k^2 \cdot 252~\implies~k^2=\frac12~\implies~k=\frac12 \cdot \sqrt{2}$

Re: figuring out height/width from area

thanks

it was that sqr root that i needed!

where do i "mark as answer"?

thanks again

Re: figuring out height/width from area

So really, you need a formula for side lengths of a given ratio; also given is area.

a = area (126 in your example), ratio = u/v (14/16 = 7/9 in your example).

x,y = side lengths

General case formula: x = SQRT(au / v) ; y = a / x

With your example:

x = SQRT(126*7/9) = 7SQRT(2)

y = 126 / 7SQRT(2) = 9SQRT(2)

Re: figuring out height/width from area

thanks

ur formula is better, as i needed it for a program script, which doesnt realy accept latex... :-)

thnak you

Re: figuring out height/width from area

Quote:

Originally Posted by

**yisman** ur formula is better, ......

For your benefit: use "your", NOT "ur".