Snowman

Calculus Level 5

Take a spherical ball of snow of radius 1 and chop off some snow from the bottom, as shown in the diagram. Re-form that snow into a smaller spherical "head" and make a snowman by stacking the head on top of the body. Measure the height of the snowman from the newly-formed flat base to the top of the head.

What is the maximum possible height of the snowman?


The answer is 2.6129.

Steven Chase by Steven Chase , 37, USA

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1 solution

Parth Sankhe
Nov 22, 2018

Here r = 1 r=1

Volume of a spherical cap = π h 2 ( r − h 3 ) πh^2(r-\frac {h}{3})

Let radius of the head be x x

4 3 π x 3 = π h 2 ( 3 − h 3 ) \frac {4}{3}πx^3=πh^2(\frac {3-h}{3})

x = ( 3 h 2 − h 3 4 ) ⅓ x=(\frac {3h^2-h^3}{4})^⅓

Total height of the body = 2 r − h + 2 x 2r-h+2x

= 2 ⅓ ( 3 h 2 − h 3 ) ⅓ + 2 − h 2^⅓(3h^2-h^3)^⅓+2-h

The maximum positive value of the above function occurs at h ≈ 0.76 h≈0.76

Hence, max value of height ≈ 2.6129 2.6129

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