Expert: Sherman D. - 11/17/2005

Question
A large birthday cake was baked to celebrate the 6th birthday of Mimi. It was a round cake, made of two layers of vanilla cake; each layer was 18 inches in diameter and 3.5 inches tall. Between the two layers was a quarter inch of chocolate frosting. And covering the top and side of the cake was 3/8" of buttercream frosting.


If the vanilla cake has 140 calories per cubic inch, and the chocolate frosting has 320 calories per cubic inch, and the buttercream frosting has 470 calories per cubic inch, how many 800-calorie servings are in the cake? Please round down to the nearest serving.


Answer
I hope you understand what i am about to tell you.

Keep in mind there are 2 layers

The 2 formulas i will use
V = pi * r^2 * h
S = 2pi * r * h

Vanilla = 2 layers
3.5 tall and 18 inches in diameter
V = 2pi * 9^2 * 3.5 = 7 * 81 * pi = 567pi in^3
140 cal/in^3 = 249379.6248 = about 249380 cal

---------------------------

Chocolate = 1 layer
.25 tall and 18 inches in diameter
Think of the thickness of the chocolate along with the length of the cake as a cylinder

V = pi * 9^2 * .25 = 20.25pi
320 cal/in^3 = 20357.5204 = about 20358 cal

-----------------------

Buttercream = 1 whole layer when you include the whole cake
3.5 * 2 layers = 7, plus .25 from the layer of chocolate plus .375 of thickness on top = 7.625 tall
(.375 on both sides becomes .75) + 18 inches from the diameter of the cake itself becomes 18.75 inches in diameter

S = 2pi * 9.375 * 7.625 = 18.75 * 7.625pi = 142.96875pi
470 cal/in^3 = 211100.3001 = about 211100 cal

211100 + 20358 + 249380 = 480838

480838/800 = about 601, 800 calorie servings

info found at http://mathforum.org/dr.math/faq/formulas/faq.cylinder.html#rccylinder

I am not 100% certain

Since word problems aren't my thing.

I guess you just have to draw it out, thats what i did. Also you have to imagine making the cake, where you cover the entire cake with buttercream except for the base of the cake.