Question

Let's suppose that you want to cook 1 lb of hot dogs using a parabolic trough concentrator on a clear June day in Los Angeles. The hot dogs would be skewered on a stiff piece of wire and aligned along the axis of the trough in the same position in which the absorber/receiver normally would be placed. The desired cooking time is 10 mins, with the resulting surface temperature of the hot dog about 450 F.
How would you design the reflector surface? Specifically, what would be the length and width of the parabolic cylinder?

Answer 1

In a parabloa of equal distance from the directrix and its focal point, I used the distance formulaand came up with this equation using generic points, with "a" being the focal distance.(d=y+a)

I then used my new equation and plotted them out on excell, and then geogebra using a focal distance of 12 inches.

On geogebra I decided what my dimentions for the cooker would be and highlighted them for a better visual representation when I started to build. I decided it would be 44 inches long and 10 inches high.

Thanks for watching‼

I found my values in this column,

subtracted the interger in this column,

I then traced a 1 inch grid on a sheet of plywood and drew my parabola, and put nails in all the points to support my styrofoam sheet.

My x axis was 1 inch off the edge of the board so my styrofoam sheet could curve arond .

origin

To start this Project I researched the definition of a parabola and found,

"The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix)."

After I traced graphs and added nails to both sides of the cooker, I added side supports and used wedges to hold my styrofoam sheet in place.

...multiplied it out

√((x-0)²+(y-a)²)=y+a

(x-0)²+(y-a)²=(y+a)²

y=x²/4a

Parabolic Hot Dog Cooker

...then I squared both sides

or

y=-p

This is my final product.

My first try cooking a hot dog, the temperature raised 10 degrees but then an overcast of clouds covered up the sun and stopped th cooking of my dinner.

Next I connected both sides. then I used 3M spray on glue to attach tin foil to the styrofoam sheet.

I tried again a few hours later after it stopped to rain and I got the weiner to start to cook. all the moisture came out and the temperature was raised to 450 F ,but then the sun started to set and the weiner stopped cooking.

Since my focal point was off my 10 inch board, I had to build extentions so I could insert my skewer, to hold the hot dogs.

and multiplied my new values by 8 so i could have my values in inches and in eigths of inches.

It also included a graphical representation simmilar to this one.

x²+y²-2ya+a²=y²+2ya+a²

...simplified and got

x²=4ya