*Balmoral Software*

Solutions: 1

Define the following notation:

R: Radius of the sphereThe volume of the spherical cone is V, where

h: Height of the spherical cap, 0 < h < R

r: Radius of the widest part, 0 < r < R

V/π = (2/3)RThe lateral surface area of the upper portion is S^{2}h

SThe lateral surface area of the lower portion is S_{cap}/π = 2Rh

SThe above three quantities are integers by convention. Then_{cone}/π = Rr

V/π = (R/3)(2Rh) = (R/3)(Sso_{cap}/π),

is rational. We have

rThe equability requirement is^{2}= R^{2}- (R - h)^{2}= 2Rh - h^{2}

The spherical cone cannot be equable if R ≤ 3, so assume R > 3 and square both sides:2h(R - 3) = 3r

4hThe coefficient of h is strictly positive and we can thus solve for h:^{2}(R - 3)^{2}= 9r^{2}= 9(2Rh - h^{2})h[4(R - 3)

^{2}+ 9] = 18R

Since R is rational, so is h. We then have

[1]

The fractional part at right has a positive numerator since R > 3. Its denominator was previously established to also be positive, so

A solution for R requires that the discriminant of this quadratic be non-negative:

4kR ^{2}- 24(k + 9)R + 45(k + 9) = 0[2]

We can then evaluate [3] to see which of the 36 possible values of k produces a square discriminant, as required by a rational solution for R in [2]. There is only one such value, and the corresponding roots for R are checked for R > 3, and in [1] to see if a valid value of h in the range 0 < h < R is produced:

Discriminant = 576(k + 9) ^{2}- 720k(k + 9) ≥ 0[3] 1 ≤ k ≤ 36

The single solution is a sphere with radius 5 and cap height 18/5, for which the volume and surface area are equal to 60 π:

k Discriminant Quadratic in R [2] Roots for R h 27 216 ^{2}108R ^{2}- 864R + 1620 = 03 (discard), 5 18/5

R = 5

h = 18/5

r = 24/5

S_{cap}= 36 π

S_{cone}= 24 π

V = 60 π

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