Math question i need help *cough* answers :)?

Question

A Sphere with a radius x-1/2 is floading in a cylindrical tank of water with a radius (x-1). Determine the volume of water in the tank. Express your answer as polynomial in factored form. Leave your answer in exact form.

Formula
PI (radius)^2 (height)-4/3 PI (radius)^3

PI(x-1)^2(x+4/6)-4/3(PI)(x-1/ 2) ^3

PLEASE HELP!!! (factor then simplify)
Thanks :)

Answer 1

Correct me if I'm wrong, but this is what you are told:
Radius of tank: x - 1
Height of tank: (x+4)/6
Radius of sphere: x - ½

That gives you this equation to simplify:
π (x-1)² (x+4)/6 - (4/3)π (x-½)^3

Hmm... let's start by multiplying out (x-1)² and (x-½)^3
(x-1)² = (x² - 2x + 1)

(x-½)^3 = (x-½)(x-½)² = (x-½)(x² - x + 1/4)
= x^3 - x² + 1/4x - ½x² + ½x - 1/8
= x^3 - (3/2)x² + (3/4)x - 1/8

This gives you:
π (x² - 2x + 1) (x+4)/6 - (4/3)π (x^3 - (3/2)x² + (3/4)x - 1/8)

Pull out the common π/3:
π/3[ (x² - 2x + 1)(x + 4)/2 - 4(x^3 - (3/2)x² + (3/4)x - 1/8) ]

Let's multiply the -4 through:
π/3[ (x² - 2x + 1)(x + 4)/2 - 4x^3 + 6x² - 3x + 2 ]

Also expand out the (x² - 2x + 1)(x + 4):
(x² - 2x + 1)(x + 4) = x^3 - 2x² + x + 4x² - 8x + 4
= x^3 + 2x² - 7x + 4

π/3[ (x^3 + 2x² - 7x + 4)/2 - 4x^3 + 6x² - 3x + 2 ]

Divide this by 2:
π/3[ ½x^3 + x² - (7/2)x + 2 - 4x^3 + 6x² - 3x + 2 ]

Combine like terms:
π/3[ -(7/2)x^3 + 7x² - (13/2)x + 4 ]

Pull out another 1/2:
π/6[ -7x^3 + 14x² - 13x + 8 ]

There might be a way to factor the polynomial, but it's not readily apparent... I'd say we are done.