Mr. Wahman and ice cream?

Question

Mr. Wahman loves to eat ice cream. He especially loves waffle cones, but he can
never get the ice cream shop to make them the way he likes them. He wants cones
which hold as much ice cream as possible, but without ice cream sticking over
the edge of the cone (too messy). He decides to design his own waffle cones. His
waffle iron makes circular waffles, which he can twist into a cone by first
making one cut along a radius. He needs to know what size cone to make.
What are the dimensions of the cone that holds the most ice cream (level to the top)?

Hint:
Cut out a circle about the size you might think a reasonable cone could be made from. Measure the radius of the circle. Draw a radius, cut it (only to the center, right! radius). Make the circle into various sized cones, measuring the new radius of its top circle. Calculate the various volumes (Volume of a Cone is 1/3 the H times the area of the circle (pi radius squared)) Make tables, draw conclusions. Remember this is real not fantasy... Cones must fit in your hand, must not collapse under the weight of the ice cream, consider what might limit your choices. Your response will not be an exact answer... more a summary of what you did and an argument as to why your response is the best possible.

Answer 1

LOL... too much warbling... the question becomes muddled.

Fix a radius R for the flat circular waffle.
Upon forming a cone from that waffle, the question boils down to determining the dimension of the cone with the maximal volume.
This time R becomes the slant height.

Let r & h be the radius and height of the cone formed.
Thus r² + h² = R²
V = (π/3) r²h
V(h) = (π/3) (R² - h²)h
V(h) = (π/3) (R²h - h³)
to optimize:
V'(h) = (π/3) (R² - 3h²) = 0 ... V'' is negative thus maximum.
h² = R²/3

h = (1/√3)R = (√3 / 3)R
r = (√2/√3)R = (√6 / 3)R
..... with these dimensions, the volume is the biggest.

Thus this is dependent on the radius of the original flat waffle... the bigger the waffle the greater the volume...



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Answer 2

This is not something we can do for you - you really have to do that one yourself.

Answer 3

huh

Answer 4

hmm...gud question...