math help algebra 1 or 2

Question

The cross section of television antenna dish is a parabola and the receiver is located at the focus, 5 feet above the vertex.

A. find an equation for the cross section of the dish.( assume the vertex is at the origin.)

b. If the television antenna is 10 feet wide how deep is it?

Answer 1

The general equation for an upward opening parabola with vertex at the origin is:

y=(1/4p)x² p is the distance from the vertex to the focus which is 5, so

the equation is y=x²/20

If you want the width of the parabola to be 10, then let x=5, as x is half the width of the parabola.

substitute and y=25/20=5/4 feet for the height or depth

A parabola is the locus of all points equidistant from the focus and directrix. The directrix is the line which is the same distance from the vertex as the focus, but on the opposite side of the focus, y=-5

For all points (x,y) on the parabola, y+5=the distance to the directrix and the distance to the focus is:

the square root of the sum of x² and (y-5)². Set those two expressions equal and square both sides

(y+5)²=x²+(y-5)² The right side is the sum of squares of the opposite and adjacent sides of a right triangle where the hypotenuse is the distance from the focus to the point (x,y) on the parabola.

Expand, simplify to get 20y=x² The y² terms cancel. The 25 terms cancel

leaving y=x²/20 as the equation for the parabola.

Check the answer, substitute (5,5/4) into y=x²/20

5/4=25/20

Answer 2

A. This is vertical parabola x^2 = 4py, where p > 0, p = 5 becouse focus above vertex. Equation of parabola x^2 = 20y.

B. If 2x = 10, then x = 5 and 5^2 = 20y, y = 25/20 = 1.25 feet