Question

I need help with this question please!
In some solar collectors, a mirror with a parabolic cross section is used to concentrate sunlight on a pipe, which is located at the focus of the mirror as shown in the diagram. What is an equation of the parabola that models the cross section of the mirror?
A.) what information can you get from the diagram?
B.) what information do you need to be able to write an equation that models the cross section of the mirror?

Answer 1

@ganeshie8

Answer 2

@jim_thompson5910 @Compassionate

Answer 3

If you choose the origin at the bottom most part, which here is the vertex, the eqtaion of the parabola is \(y = ax^2\).
The focus for such a parabola will be \(\large (0, \frac{1}{4a})\)
1/4a = 6. a = 1/24
\(\large y = \frac{1}{24}x^2\)

Answer 4

Is that it?

Answer 5

Yes, as far as I can tell.

Answer 6

The questions are somewhat backwards. First they ask: What is an equation of the parabola? Then they ask two more questions A and B.

Answer 7

A) what information can you get from the diagram?
The parabola has a vertical axis and it opens upward which means the equation of the parabola is of the form y = a(x-h)^2 + k where a is positive and (h,k) is the vertex.
We can also get the information the focus is 6 feet above the vertex.

Answer 8

B.) what information do you need to be able to write an equation that models the cross section of the mirror?
We need to know the location of the origin of the x-y coordinate system.
If we make the assumption the origin is at the vertex, then, the location of the vertex (h,k) is (0,0) and the equation of the parabola reduces to y = a(x-0)^2 + 0 or
y = ax^2.
The location of the focus from the diagram is (0, 6).
1/(4a) = 6. a = 1/24. y = 1/(24) * x^2 is the equation of the parabola.