Please help me with this question!?

Question

A craft company wants to develop and market decorative Styrofoam snowmen. After some discussion with the company executives about the proper proportion of head and body, a decision was made to use two spheres. The larger body sphere will be 1.6 times the diameter of the smaller sphere (head). The base of the body sphere will be cut off at 0.8 of its radius. The base of the smaller sphere will be cut off at the neck at 0.9 of its radius to fit exactly with a matching cut at the top of the body sphere (the two radii of the cut surfaces of the two spheres will be exactly equal).
Derive an equation in terms of the radius of the head (r) that will allow you to calculate the volume of material required for manufacturing the snowmen. Simplify to a minimum number of terms.

Answer 1

To solve this problem, you need to know how to calculate the volume of a region of a sphere lying above a specified plane. Such a region is called a "spherical cap". If the radius of the sphere is "R", and the plane cuts the sphere such that the height of the cap above the plane is "h", and the radius of the circle formed by the intersection of the plane with the sphere's suface is "a", then the volume of the cap is given by:

V = (pi/3)* h^2 * (3*R - h)

We also have, from the Pythagorean theorem, that:

a^2 + (R-h)^2 = R^2

See source for a figure and the origin of these formulas.

We will use both of these formulas.

Let's start with the head. Let the radius of the sphere forming the head be "r". We are told that the base of the head is a plane cutting the sphere at a distance 0.9r from the center, so in this case, h = r/10. The volume of the head is given by the volume of the intact sphere minus the volume of the missing spherical cap:

V_head = 4/3 * pi * r^3 - 1/3 * pi * (r/10)^2 * (3*r - r/10)
V_head = (4*pi*r^3)/3 * - (pi*r^3)/100 + (pi*r^3)/3000
V_head = (3971/3000)*pi*r^3

Now, we need to calculate the radius of the circle forming the "neck" of the snowman. This is the parameter "a" in the above formulas. We can then use that value to calculate "h" for the missing spherical cap on the body of the snowman, where the head meets the body.

Using the second formula above, we have that:

a^2 = r^2 - (r - r/10)^2

a^2 = (19*r^2)/100

a = sqrt(19)*r/10

We now need to solve the second "cap" formula above for h in terms of a and R.

a^2 + (R-h)^2 = R^2
h^2 - 2Rh + a^2 = 0

Using the quadratic formula:

h = R +/- sqrt(R^2 - a^2)

the square root term is positive, and h must be less than R, so the only root that makes sense physically is the one involving the minus sign:

h = R - sqrt (R^2 - a^2)

Now use the value we calculated above for a in this formula to calculate h for the body. We are told that the radius of the sphere forming the body is 1.6*r, and we'll use that to express the radius of the body sphere in terms of the radius of the head sphere:

h = 16r/10 - sqrt((16r/10)^2 - (19*r^2)/100)
h = 16r/10 - sqrt((237r^2)/100)
h = (16 - sqrt(237))*r/10

We are now ready to calculate the volume of the partial sphere forming the body. Again, the volume is the volume of the "full" sphere with a radius of 16r/10 minus the volume of the spherical cap at the top (with h given by the value we just calculated), and the bottom (with h = 0.2 times the radius of the large sphere, which is equal to 16r/50 in terms of the radius of the "head" sphere.).

V_body = 4/3 * pi * (16r/10)^3 - 1/3 * pi * (16r/50)^2 * (3*(16r/10) - 16r/50) - 1/3 * pi * ((16 - sqrt(237))*r/10)^2 * (3*(16r/10) - (16 - sqrt(237))*r/10)

after some tedious algebra (which you should check!), we have that:

V_body = pi * r^3 * (120832/46875 + 177*sqrt(237)/1000)

The total volume is given by the volume of the head plus that of the body:

V = V_head + V_body

V = (3971/3000)*pi*r^3 + pi * r^3 * (120832/46875 + 177*sqrt(237)/1000)

V = [487677/125000 + 177*sqrt(237)/1000] * pi * r^3

Numerically, (to 7 significant figures) this is equal to:

20.81712 * r^3

Answer 2

It's a hairy derivation. I found these pages on the Doctor Math website. Ref.1 gives the volume equation for the "cap" (truncated portion) of the sphere and ref. 2 has the derivation. Application to your problem shouldn't require much more that some trig functions and scaling.
Good luck.