Don't understand how they figured out the equations.

So I have attached the problem that is part of my final review. I have also the code that will give me the answer (shown below). My question is... how did they figure out the equation for "L" and "P"?? I know that it is mostly algebra but still not sure. Any help would be great!

w = 6;  %width in meters
A = 80; %area in meters squared
%solve for the length, terms of A, w
L = (A - w^2/4)/w;
%solve for the perimeter
P = 2*L + w + 2*w/sqrt(2);
%display your answers
fprintf('The length is %5.2f\n ', L)
fprintf('The total perimeter is %5.2f\n ', P)
%done

Answer 1

[the cyclist]

Here are some facts that may help you make the connection:

The enclosure (area = A) is made up of two parts:

• A rectangle of area L * W
• A triangle of area D*D/2

The triangle is a right triangle with side lengths D, D, and W. W is the hypotenuse, so

• D^2 + D^2 = W^2

Is that enough for you to see how the whole thing works?

Answer 2

[Star Strider]

I don’t see ‘P’ anywhere, so I’ll wait for clarification on it.

The rest is relatively straightforward:

First, consider the triangle on the right —

W^2 = 2*D^2
D = W/sqrt(2)

For the area, since it’s a right triangle, D = base = height and the area of it is —

Atri = 0.5 * D^2

The area of the rectangle bounded by L and W is obvious.

So the area of the entire figure is —

Atot = W * L + 0.5 * D^2

I’ll leave you to simplify this expression and do all the coding. (It’s all just secondary-school geometry and algebra.)

[the cyclist]

P is the perimeter.

[Star Strider]

I assumed that, but since it was over the horizon (not in the Question), I wanted to be sure. I also wanted to be sure it wasn’t ‘D’.

Answer 3

[Jessica]

ok I see. I was confused on why they were dividing by 4 to find "L". Thanks for the help!! :)