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Rational root theorem question!?
The Universal Paper Product Company makes cone-shaped drinking cups. the height of each cup os 6 cm more than the radius. If the volume of each cup is 27π cubic cm, what are the dimensions of the cup?
(1/3)π r^2h = 27π
r^2(r + 6) = 81
r^3 + 6r^2 - 81 = 0
Now the rational roots theorem tells us that the only rational numbers that could work as solutions must divide 81. We try various divisors and it turns out that 3 works.
3^3 + 6*3^2 = 27 + 54 = 81.
Do any other numbers work? Divide the polynomial by r - 3.
r^3 + 6r^2 - 81 = (r - 3)(r^2 + 9r + 27)
The second factor, r^2 + 9r + 27, has no real roots, because b^2 - 4ac is negative. So 3 is the only solution to the problem. So the radius is 3 cm and the height is 9 cm.
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