Solve Calculus & Conics: Parabolic Reflector, Opening 10cm, Diameter 7cm

[bmr676]

Homework Statement

A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. Find an equation of the parabola where V is at the origin. Find the diameter of the opening | CD |, 7 cm from the vertex.

Because the picture is not on here as well, it should be mentioned that the focus is closer to the vertex than 7cm. Also, it is labeled that the 10cm opening at the focus is from point A to B.

Homework Equations

x=ay²

The Attempt at a Solution

At the focus, the coordinates are (x,5) and (x,-5) for A and B respectively. Vertex coordinates are (0,0). Equation is x=a(y^2), thus at the focus, the equation is x=25a. Also, the points C and D are (7,y) and (7,-y) respectively. So the equations for those would be 7=a(y^2).

I feel that not enough information was not given with this problem, but I must be missing something. Any inputs?

 

[mighty2000]

Hello, thank you for your question. From the given information, we can determine that the parabola has a vertical axis of symmetry since the focus is located on the y-axis. We also know that the opening at the focus is 10 cm, which means that the distance from the focus to the vertex is also 10 cm. This gives us the value of a, which is 1/20.

Now, to find the equation of the parabola, we can use the standard form of a parabola, which is y^2 = 4ax. Substituting the value of a, we get the equation y^2 = (1/5)x.

To find the diameter of the opening |CD|, we can use the distance formula between the points C and D. This gives us the equation (7 - 0)^2 + (y - 0)^2 = (7 - 0)^2 + (-y - 0)^2. Simplifying this, we get y^2 = 49/5. Now, we know that the diameter is twice the value of y, so the diameter is 2(sqrt(49/5)) = 14/√5 ≈ 6.26 cm.

I hope this helps. Let me know if you have any further questions.