Lenny Conundrum #180

[AySz88]

I saw this at least an hour ago, and I'm surprised nobody posted it yet...

To promote a new delivery service, Pizzaroo is including a free Neopets Collectable Card with every pizza purchase. What a deal! So you decided to order a Jelly Bean Pizza. Yum!



When the pizza arrived, you noticed that the free Neopets Collectable Card was in the corner of the square box, so that its corner just barely touched the outside of the perfectly-circular pizza. The Collectable Card measures exactly 140 mm by 70 mm.

What is the diameter of the pizza, in millimetres? Please round to the nearest millimetre.



P.S. Unfortunately, because of the difficulty in delivering a jelly bean pizza without all of the jelly beans falling off, the pizza delivery service was discontinued soon after it began.

Sum of digits of radius = 8

[kimiko_00]

Sum of digits is 8?
I got a sum of 7..

[ana_lucia]

The sum of the digits of the radius is 8, but the answer was the diameter...

[AySz88]

Sum of digits is 8?
I got a sum of 7..

The sum of the digits of the radius is 8, but the answer was the diameter...

Sorry for the roundaboutness, but I had a reason.... (I won't spill the beans yet though.)

[Shoyru_Lover]

Quite simple really. The answer is Pie.

[miz sparrow]

I hate geometry.

hate hate hate.
give me algebra any day!

[AySz88]

Sum of digits is 8?
I got a sum of 7..

The sum of the digits of the radius is 8, but the answer was the diameter...

Sorry for the roundaboutness, but I had a reason.... (I won't spill the beans yet though.)

Okay, the reason for that was that 700 is somewhat easy to randomly guess when you get "the sum of the digits of the answer is 7". Hopefully, random-guessers tried radius 8 or 80 (diameter 16 or 160) after the additional hints instead of 350 and 700.

Since the prize message is out, here is the solution:

Let's put this circle on a coordinate system, with the origin in the center of the circle. We would need one more point to find the radius, and we have it.

Let's consider where the pizza box is - it creates a square around the pizza. Concentrating only in the upper-right, the box completes a square with side r around the 1/4-slice of pizza (use the x-axis, the y-axis, and two sides of the box). So one corner of the card touches (r, r) against the corner of the box. The other corner must, then, touch (r-140, r-70), or vice versa (it doesn't matter).

With this information, we would have to solve the following equation:

x^2 + y^2 = r^2 such that (r-140, r-70) is a point on the circle.

Substitute the point into the equation.

(r-140)^2 + (r-70)^2 = r^2
...
r^2 - 420 r + 24500 = 0
(r-70)(r-350) = 0

Solving for r, you get the two solutions r = 70 or r = 350.

r = 70 is too small (the card would take up half of the box), so that one is rejected.
r = 350 is only radius remaining
The diameter is 700.

[Mirry]

I just guessed 700, because my thumb was the same length as the card (sort of)
And 5 thumbs was the diameter.
5*120 = 700

And i won 427 np. Hooray!