Est. 1998 — 27 years of woodworking knowledge
Pizza Day Extra
The 'Pies Aren't Square' Pizza oven company manufactures a line of wood-fired pizza ovens. All of the ovens have square hearths. Model numbers are based on the capacity of the oven.
For example, the Model #1 is designed to bake one 16" round pizza and has a 16" square hearth. The Model #4 is designed for four 16" round pizzas and has a 32" square hearth.
What is the size of the hearth for the Models #2, #3 and #5?
With or Without Pepperoni??
Re: Pizza Day Extra
think hypotenuse
Re: Pizza Day Extra
If measured to the nearest 16th of an inch, are the fractional parts:
#2 5/16"
#3 7/16"
#5 5/8"?
And even if right, is there a way to get #3 without trig?
Your hint
is consistent with my solution for #2 and #5. But I can't figure #3 without resorting to trig. Maybe some long-forgotten (at least by me!) half-angle formula?
Re: Pizza Day Extra
Your fractional parts are good.
I've seen an answer for #3 expressed in non triginometric terms, but I used trig to get there myself.
Black olives and extra anchovies
Re: Pizza Day Extra
Answer:
Model #2 - 27.312"
Model #3 - 31.448"
Model #4 - 38.624"
The general formula for unit circles (r=1) in a square is:
#2: 2 + �sqrt(2)
#3: 2 + 1/sqrt(2) + �sqrt(6)/2
#5: 2 + 2�sqrt(2)
Alex, I'd be interested to see you trig solution for #3.
#3 with trig
Here's my solution for #3, showing the square to be 16 + 16*cos(15 degrees).
Am I the only one whose screen shows funky symbols where I think you typed "2/"?
Let's see your nontrig solution to #3.
diagram
too quick on the "post" button!
Re: #3 with trig *LINK*
Actually Alex, my first approach to all of them was with trig. My solution to #3 was pretty much the same as yours: 16 + 16*sin(75).
The link below shows answers that appear to be pythagorean in nature, but with no derivation. 2 and 5 are pretty straightforward to derive. I didn't have any spare time over the weekend to work on 3.
http://www.stetson.edu/~efriedma/cirinsqu/
#3 nontrig
Okay, had a chance to figure this one out:
I created a "path" from one vertical edge to the opposite, as shown by the bold lines below.
Two segments are just the radii of two circles that those edges are tangent to. 16
Two segments use the equilateral 16" triangle among the centers of the three pizzas The altitude of that triangle is 8*sqrt(3), and half the base is 8. Since both those segments are along lines at 45 degrees , we divide by sqrt(2) to get the horizontal components of those distances. 8*(1+sqrt(3))/sqrt(2)
Nicely done!
Re: Nicely done!
Thanks. Hopefully no one is confused by the error in my drawing at lower left corner. That obviously should be an "8", not "8/sqrt(2)"