Help

[hasty ocean]

solving for these constraints yields a cubic 12x^3 + 36x^2 = 76.8^2, which can be solved using approximation

or the cubic formula if you're brave lol

I got that cubic by combining 1/2xy=38.4 and x^2+y^2=(x+6)^2

[hasty ocean]

u know how 1/2bh is the equation for area of a triangle, and a^2+b^2=c^2 is the equation relating sides of a right angle triangle?

[brazen zealot]

.close other leg's length is (√12) (∛(√122.88 + √121.88) + ∛(√122.88 - √121.88))

have (x, y) represent a right triangle with one of the legs being x and the other leg being y

hypotenuse is 6 inches longer than one of the sides
=> sqrt(x^2 + y^2) = 6 + x
=> x^2 + y^2 = (6 + x)^2
=> x^2 + y^2 = x^2 + 12x + 36
=> y^2 = 12x + 36

area is 38.4 square inches
=> xy/2 = 38.4
=> xy = 76.8

y^2 = 12x + 36
y^3 = 12xy + 36y
y^3 = 12(76.8) + 36y
y^3 - 36y = 921.6
y(y^2 - 36) = 921.6
y(y + 6)(y - 6) = 921.6
y > 6

3y^2 - 36 = 0
y^2 = 12
y = +-sqrt(12)
at y = -sqrt(12), y(y^2 - 36) = (-sqrt(12))((-sqrt(12))^2 - 36) = 48 sqrt(3) < 921.6
so one positive solution with y > 6

let a = sqrt(12), so that a^2 - 12 = 0
3y^2 - 36 = 0 => y^2 = 12 => y = +-sqrt(12)
replace y with a(t + 1/t)
(a(t + 1/t))^3 - 36a(t + 1/t) = 921.6
a^3(t^3 + 3t + 3/t + 1/t^3) - 36a(t + 1/t) = 921.6
a^3(t^3 + 1/t^3) + (3a^3 - 36a)(t + 1/t) = 921.6
a^3(t^3 + 1/t^3) + 3a(a^2 - 12)(t + 1/t) = 921.6
12 sqrt(12) (t^3 + 1/t^3) = 921.6
144 (t^3 + 1/t^3) = 921.6 sqrt(12)
(t^3 + 1/t^3) = 6.4 sqrt(12)
t^3 + 1/t^3 = 12.8 sqrt(3)

t^3 + 1/t^3 = 921.6 / (12 sqrt(12))
t^3 + 1/t^3 = 12.8 sqrt(3)
t^6 + 1 = 12.8 sqrt(3) * t^3
t^6 - 12.8 sqrt(3) * t^3 + 1 = 0

t^3 = (12.8 sqrt(3) +- sqrt((12.8 sqrt(3))^2 - 4(1)(1))) / 2
t^3 = 6.4 sqrt(3) +- sqrt(121.88)
t^3 = sqrt(122.88) +- sqrt(121.88)
- case would mean t < 1, so choose + case
t^3 = sqrt(122.88) + sqrt(121.88)
t = cbrt(sqrt(122.88) + sqrt(121.88))

notice that (sqrt(122.88) + sqrt(121.88))(sqrt(122.88) - sqrt(121.88)) = 122.88 - 121.88 = 1
so the + and - cases are reciprocals of each other
1/t = cbrt(sqrt(122.88) - sqrt(121.88))

y = a(t + 1/t)
y = sqrt(12)(cbrt(sqrt(122.88) + sqrt(121.88)) + cbrt(sqrt(122.88) - sqrt(121.88)))