+113 If the height of an equilateral triangle is also a root of the equation y4-3y2-270, then the area of the triangle, in cm2 is,
A) \(8√2\)
B) \(6√3\)
C)\(12√2\)
D)\(9√3\)
E)\(3√6\)
+128448 y^4 - 3y^2 - 270 = 0 factor as
(y^2 - 18) (y^2 + 15) = 0
Only the first factor will result in a real (positive) answer
So
x^2 - 18 = 0
x^2 = 18 take the positive root
x = √18
√3 = x / (3/√2)
And because we have an equilateral triangle we can find (1/2) base ,x, thusly
tan 60 = √18/ x ⇒ x = √18/tan (60)
So.....the area is
(1/2)base * (height) =
√18 * √18 / tan (60) =
18 / √3 =
18√3/ 3 =
6√3 units^2
