math help!!! Please

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

Since this problem is about a satellite antenna dish, I would first assume that the dish is pointing upwards towards the sky. If we were to take a cross section (a vertical slice of the dish) it would be a parabola opening upwards.

For a parabola opening upwards, the equation is in the form:

(x–h)² = 4p(y–k)

Here, (h, k) is the vertex and since the problem says to put the vertex at the origin (0,0) both h and k = 0.

This simplifies our equation to:

x² = 4py

Our only unknown is p.

We can solve for p using the information given about the focus. The focus is a point within the opening of the parabola along the axis of symmetry. Since the focus is 5 ft above the vertex, this would give coordinates of (0,5) for the focus.

The focus is also found to be: (h, k+p)

Substituting our h and k values we get (0, p). Since we have determined that the focus is at (0, 5) p must be equal to 5. This allows us to rewrite our equation as:

x² = 4(5)y

or

x² = 20y FINAL ANSWER

Now for part b.

Since the satellite dish is 10 ft wide, if we were to imagine at parabola graphed with the vertex at the origin, a width of 10 would be split into 2 halves of 5 units on each side of the y-axis since parabolas are symmetric. This would make the coordinates of this location (-5, y) and (5, y). We just don't know what the y value is at this point.

If we take 5 (or -5) and substitute it in for x in our equation of the parabola, this would let us solve for y, which would be the height of dish where the full width is 10 ft.

5² = 20y

y=25/20

y=1.25 ft. FINAL ANSWER

Hope this explanation helps!