Franz and the Cake Fitting
10,000 = A = R*R
The area of the circular layer will have to be 10,000 cm squared.
R = 56.42 cm.
Thus, the diameter of the circle, which will be equivalent to the side length of the square box, is 112.84 cm. To the nearest decimal place, the square box will have dimensions of 112.8 cm x 112.8 cm.
10,000 = A = R*R
This week's top honors go to Dean Vendramin [firstname.lastname@example.org]. He executed the sauteing and simmering of this tasty dish to a perfection. Magnifico, Dean!
Knowing area formulas, solving equations techniques, and parts of a circle are critical to solve this problem.
The area of the square cake would be:
A = l * w
100cm * 100cm = 10000 cm^2
The couple agreed that the area must stay the same but now it should be a circle. Using the area of a circle formula (A = pi * r^2) (pi~3.14), we can find the radius of the circle by using the following problem solving method:
10000cm^2 = 3.14 * r^2 (now divide both sides by 3.14)
3184.7cm^2 = r^2 (now take the square root of both sides)
56.4 cm = r
The radius of the cake is about 56.4 cm.
But the radius only gives us the distance from the middle of the circle to the edge, so we must find the diameter of the circle. Using the formula D = 2r, the diameter of this circle is D = 2(56.4cm) or D = 112.8 cm.
So the smallest box that Franz could use would have the length and width of 112.8 cm.
Minus' buddy, HaL0N, is back with yet another delectable wonder. Our mystery chef sure knows how to whip-up some steamy hot food. Won't you please tell us about yourself, HaL0N?
Solution to Franz and the Cake Fitting (brought to you by HaL0N):
Since the dimensions of the original top layer are already known (100 cm x 100 cm), all that is needed is the radius of the new circular layer. The area of the new layer must be the same as that of the square layer (10,000 square cm).
So, the radius is found by solving for it in the equation, A = Pi * r^2, where 'A' denotes the area and 'r' denotes the radius. After simple calculations, the radius is found to be 56.419 cm. Since Franz is looking for the smallest possible box to fit the cake, the sides of the box should be as large as the diameter of the cake. The diameter of the cake is 2r. Therefore, the sides of the box should measure 112.8 cm x 112.8 cm.
Michael Smith [email@example.com] makes a valid point about cakes...they are 3-dimensional. Well, at least the good ones are. The cakes we make around here always turn out a little flat :P (yuk!)
The dimensions of the box are 112.8 cm by 112.8 cm, but without knowing the height of the cake, it is impossible to determine the exact dimensions of the box, because I'm sure the cake wasn't two dimensional. If the box, however, was in the shape of a cube, the third dimension of the box would also be 112.8 cm because cubes have the same height, width, and length.
Our budding feminist, Emma Rohrlach from Methodist Ladies' College in Claremont, Western Australia, ended up with a really GIGANTIC wedding cake in her solution. She didn't get the right answer, but she gets top marks for keeping our mascot in line.
Before I give you my solution I want to say how sexist and unfair this week's problem is. Why should the groom get to choose that the cake is circular, after all the bride's father usually has to pay for it!!!!
Anyway, here's my solution:
If the cake's top is the same area as the rectangular one then it will be 10,000 cm squared. To find the area the formula pi x 2radius or pi x diamtere is used so to work out the diameter of the cake we divide 10,000 by pi. This gives 3183.099 cm as the diamter. If the box is touching the sides lightly without squishing the cake then this will also be the length of one side of the box. So to one decimal place the dimensions would be 3183.1 cm x 3183.1 cm.
Next time I think the female should have a bigger input, don't you??? Until we meet again!