Can anyone solve this, i need the explination not just the answer?

Question

When the pizza arrived, you noticed that the free 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.

Answer 1

it depends on whether the pizza is round, and if I ate some or not

Answer 2

Call the radius of the pizza 'a' millimetres. The coordinates of the point where the corner of the card touches the pizza are (with the pizza centre as origin): (a-70, a-140). Therefore
(a-70)^2 + (a-140)^2 = a^2. Expanding the squares and collecting all terms leads to the quadratic equation
a^2 - 420a + 24500 = 0.
The solutions are (1) a = 70, which is not the one we want as it corresponds to the card covering the top half of the box and both the lower corners touch the pizza at the ends of a diameter; and (2) a = 350, which is the only correct solution to the problem.

Answer 3

Assume
- the pizza is a perfect circle
- the box is a square tangent to the circle

Therefore
- the diagonal of the collectors card is touching the square at one point and the circle at another.
- for each pizza diameter there is one and only one box size.

Notation:
i - hat, the unit vector in the x direction and j-hat, the unit vector in they direction are long to type.. the symbols used will be i and j respectively.

I would attempt
xi + yj + 140mm i + 70 mm j = d/2 i + d/2 j
sqrt(xi^2 + yj^2) = d/2

or due to independence of vectors
xi + 140mm i = d/2 i
yj + 70 mm j = d/2 j
sqrt(xi^2 + yj^2) =d/2

thats three equations and three unknowns.

You can solve for d and the answer is valid.
They didnt tell you whether the card was vertical or horizontal so you cant be sure you have the right x and y values.

make sure your units are given in mm so they are consistent.

Now I have given you a way to solve it, whithout solving it for you. How does that work for you.

Answer 4

Suppose the center of the pizza is at (0, 0), and its radius is r.
The equation of the pizza is x² + y² = r².
The pizza touches the point (r - 140, r - 70). [One corner of the pizza box is (r, r), and the card "comes back" from that point 140 in one direction and 70 in the other.]

Substituting,
(r - 140)² + (r - 70)² = r²
r² - 280r + 19600 + r² - 140r + 4900 = r²
r² - 420r + 24500 = 0
(r - 70)(r - 350) = 0
r = 70 or r = 350

If r = 70, then the entire pizza covers the 140 × 70mm card.

The radius is 350 mm... pretty big pizza!