A wire of length L is cut into two pieces of length X and Y respectively, with Y>X. These pieces are then used to form an upper case "L". If Y/X= (X+Y)/Y, determine the values of X and Y in terms of L.
y/x = (x+y)/y
since x+y=L,
y/x = L/y
Lx = y^2
Lx = (L-x)^2
x^2 - 3Lx + L^2 = 0
now solve as usual for quadratic. I expect you will get only one usable value
Similarly,
Lx = y^2
L(L-y) = y^2
y^2 + Ly - L = 0
now do the same for y.
Make sure you choose positive values such that x+y=L
To solve this problem, we need to express Y and X in terms of L based on the given equation.
Given:
Y/X = (X+Y)/Y
We can start by cross multiplying the equation:
Y * Y = X * (X + Y)
Expanding:
Y^2 = X^2 + XY
Next, let's isolate XY on one side of the equation:
XY - X^2 = Y^2
Now, we can express X in terms of Y:
X = Y^2 / (Y - X)
Similarly, we can express Y in terms of X:
Y = X^2 / (Y - X)
To relate X and Y to L, we know that the sum of X and Y is equal to the length of the original wire L:
X + Y = L
From here, we can substitute the value of Y in terms of X into the equation X + Y = L:
X + X^2 / (Y - X) = L
Simplifying this equation will give us the value of X.