an engineer designs a satellite dish with a parabolic cross section. The dish is 15 ft wide at the opening and the focus is placed 4 ft from the vertex. find an equation of the parabola.
I know how to work this problem, but how do I know that I use the equation y^2 = 4px instead of x^2 = 4py.
Please help this is so confusing!!!!
Notice the wording: Find AN equation of
the parabola. I believe the word AN
implies that there is more than one possibility.
If there was only one possibility, the problem would have said: Find THE eguation of the parabola. So I believe an X or Y-parabola is correct.
Ah, the great conundrum of parabolic equations! Don't worry, my friend, I'm here to bring some humor to the mix and help you out.
Think of it this way: you have a satellite dish, which is basically a big ol' bowl. Now, if you were to pour soup into it, would it spill out of the x-axis or the y-axis?
I bet you were tempted to say the x-axis because that's where the opening is, but that's the sneaky part! The soup (or focus, in this case) is placed at the bottom of the dish, right? And where does the soup collect at the bottom? Well, down the y-axis, of course!
So, we use the equation y^2 = 4px because it represents the parabolic cross section correctly. Parabolas open up or down, just like the soup would go down along the y-axis in this dish. It's all about choosing the right direction and keeping your soup (or focus) in the right place!
Now, go forth and solve that parabola equation like a pro, my friend! Soup's on!
To determine whether to use the equation y^2 = 4px or x^2 = 4py, you need to analyze the given information about the satellite dish.
In this case, the dish is 15 ft wide at the opening, which means the length of the opening across the x-axis is given by 2a = 15 ft, and thus a = 7.5 ft.
Since the focus is placed 4 ft from the vertex, the distance from the focus to the vertex (p) is 4 ft.
With these values, you can now determine the correct equation.
1. If the dish has a "parabolic cross-section opening towards the x-axis" (width-wise), then the equation is x^2 = 4py.
2. If the dish has a "parabolic cross-section opening towards the y-axis" (height-wise), then the equation is y^2 = 4px.
In this case, we know that the dish is 15 ft wide at the opening, which means the opening is width-wise along the x-axis. Therefore, we need to use the equation x^2 = 4py.
Hope this clarifies the confusion!
To determine whether to use the equation y^2 = 4px or x^2 = 4py for a parabola, you need to consider the orientation of its cross-section. In this case, we have a satellite dish with a parabolic cross-section.
First, let's understand the equation y^2 = 4px. This equation represents a parabola with the focus located on the positive x-axis. The vertex of the parabola is at the origin (0,0), and the p-value determines the distance from the vertex to the focus.
Now, let's consider x^2 = 4py. This equation represents a parabola with the focus located on the positive y-axis. The vertex is also at the origin, and the p-value determines the distance from the vertex to the focus.
In the case of a satellite dish, the focus is placed 4 ft from the vertex. Since the dish is 15 ft wide at the opening, we know that the dish opens along the x-axis (horizontal opening). Therefore, we'll use the equation y^2 = 4px.
For the given problem, the focus is 4 ft from the vertex. We can determine the value of p by using the opening width, which is 15 ft. The distance from the focus to the directrix is equal to the distance from the directrix to the vertex, which is p.
In this case, the width of the dish at the opening is 15 ft, so the distance from the vertex to the opening is half of that, which is 7.5 ft. Therefore, p = 4 ft + 7.5 ft = 11.5 ft.
Now, we can use the equation y^2 = 4px with the value of p we found. The equation becomes y^2 = 4(11.5)x, which simplifies to y^2 = 46x.
Therefore, the equation of the parabola that represents the cross-section of the satellite dish is y^2 = 46x.