Respuesta :
The telescope shape and the characteristic equations of the telescope parameters are the same as parabolic equations
The distance between the focus and the vertex, of the parabola is 3.375 meters
The process for obtain the above values is as follows:
The known parameters of the parabola are;
The location of the vertex of the parabola= The origin = (0, 0)
The height of the parabola = 9 meters
The width of the parabola = 12 meters
The unknown parameter;
The distance between the focus and the vertex
Method:
Finding the coordinate of the focus from the general equations of the the parameters of a parabola
The equation of the parabola in standard form is y = a·(x - h)² + k
From which we have;
(x - h)² = 4·p·(y - k)
The coordinates of the focus, f = (h, k + p)
Where;
(h, k) = The coordinates of the vertex of the parabola = (0, 0)
∴ a = 1/(4·p)
From the question, we have the following two points on the parabola,
given that the parabola is 12 meters wide at 9 meters above the origin and
it is symmetric about the y-axis;
Points on the parabola = (9, 6), and (9, -6)
Plugging in the values of the vertex, (h, k) and the two known points, in the equation, y = a·(x - h)² + k, we get;
6 = a·(9 - 0)² + 0 = 81·a
a = 6/81 = 2/27
p = 1/(4·a)
∴ p = 1/(4 × 2/27) = 27/8
The coordinate of the focus, f = (h, k + p)
∴ f = (0, 0 + 27/8) = (0, 27/8)
The coordinate of the focus f = (0, 27/8)
Given the vertex and the focus of the parabola have the same x-values of 0, we have;
The distance between the focus and the vertex, d = the difference in their y-values;
∴ d = 27/8 - 0 = 27/8 = 3.375
The distance between the focus and the vertex, d = 3.375 meters
Learn more about parabola here;
https://brainly.com/question/22404310
