Q.3. Simultaneous equations in 3 unknowns by the famous mathematician Abu Kamil

Question 3:

Abu Kamil was a famous Arab mathematician who worked in the field of algebra. (See en.wikipedia.org/wiki/Abu_Kamil). This is one of his famous problems. It is rather simple and only needs the algebra learned in the first year of high school. Yet, I have posed this question to a graduate student in math, a junior in math, a PhD biomedical engineer, a high school student, and the Yahoo math section. Nobody has given me the correct answer. I'm trying again.

x^2 + y^2 = z^2

y^2 = xz

x + y + z = 10

x, y and z are all positive real numbers. They are approximately 2, 3 and 4. What are they to 5 significant digits?

Hint, substitute the second equation into the first and use the formula for finding the root of a quadratic equation.

Answer 3:

x = 2.5706,

y = 3.2699

z = 4.1594

worked out as under.

Let x = rcosθ and y = rsinθ

=> r^2cos^2 θ + r^2sin^2 θ = z^2

=>
z = r

y^2 = zx
=> r^2sin^2 θ = r^2 cosθ
=> sin^2 θ = cos θ
=> cos^2 θ + cos θ - 1 = 0
=> cos θ = (1/2)(-1 + √5) = 0.61803
=> sin θ = √[1 - (0.61803)^2] = 0.78615

x + y + z = 10
=> r(cos θ + sin θ + 1) = 10
=> r = 10 / (1 + 0.61803 + 0.78615) = 4.15942
=>
x = r cos θ = (4.15942)*(0.61803) = 2.5706
y = r sin θ = (4.15942) * (0.78615) = 3.2699
z = r = 4.1594.

Link to YA!