Answer:
The diameter at the center of the tower is 4 meters. The center of the tower is 8 meters above the ground.
Explanation:
4x^2-y^2+16y-80=0
Completing squares in variable "y": Common factor -1:
4x^2-(y^2-16y)-80=0
4x^2-[(y-16/2)^2-(16/2)^2]-80=0
4x^2-[(y-8)^2-(8)^2]-80=0
4x^2-[(y-8)^2-64]-80=0
Eliminating the brackets:
4x^2-(y-8)^2+64-80=0
Adding like terms (constants):
4x^2-(y-8)^2-16=0
Adding 16 both sides of the equation:
4x^2-(y-8)^2-16+16=0+16
4x^2-(y-8)^2=16
Dividing all the terms by 16:
4x^2/16-(y-8)^2/16=16/16
Simplifying:
x^2/4-(y-8)^2/16=1
The hyperbola has the form:
(x-h)^2/a^2-(y-k)^2/b^2=1
Then:
h=0
k=8
a^2=4→sqrt(a^2)=sqrt(4)→a=2
The diameter (d) at the center of the tower is:
d=2a→d=2(2)→d=4 meters
The center of the tower is 8 (k) meters above the ground.