Algebra 2

You are designing a kit for
making sand castles. You want one of the molds to
be a cone that will hold 48π cubic inches of sand.
What should the dimensions of the cone be if you
want the height to be 5 inches more than the radius
of the base?

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  1. let the radius be x inches
    then the height is x+5
    V of cone = (1/3)πr^2 h
    48π = (1/3)π x^2 (x+5)
    multiply by 3/π

    144 = x^3 + 5x^2
    x^3 + 5x^2 - 144 = 0
    try only positive values of x, and factors of 144
    clearly you can work in out mentally that x = 1, 2 and 3 are too small
    try x = 4
    64 + 80 - 144 = 0 , yeahhh! , so x-4 is a factor

    by synthetic division
    x^3 + 5x^2 - 144 = (x-4)(x^2 + 9x + 36) = 0
    the remaining quadratic only has complex roots, so
    x = 4
    So the cone has a radius of 4 and a height of 9 inches

    Proof:
    V = (1/3)π(16)(9) = 48π

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