Originally Posted by

**jonnygill** I'm sorry, in the initial problem i said he has a cube of ice. He in fact has a three dimensional rectangular figure.

This problem is broken up into 4 parts: a, b, c, and d.

All parts concern a sculptor who has a block of ice that measures 3 x 4 x 5 = 60 ft^3. Before he starts sculpting, he wants to shave off the same number of feet from each dimension to make the block of ice smaller.

a asks to write a polynomial function to model the situation. I came up with $\displaystyle V(x)=-x^3+12x^2-37x+60$ This function shows Volume in terms of the amount removed from each dimension, x. The function itself was derived from (3-x)(4-x)(5-x) where x is the number of feet removed from each dimension.

b asks to graph the function (graphing calculator)

c states that he wants to reduce the volume TO 3/5 the original volume and to write an equation to model this situation... I figured that if $\displaystyle V(x)=60=-x^3+12x^2-37x+60$ then $\displaystyle \dfrac{3}{5}\cdot60=-x^3+12x^2-37x+60$ or $\displaystyle 36=-x^3+12x^2-37x+60$ or $\displaystyle 0=-x^3+12x^2-37x+24$

d asks how much he should remove from each dimension... essentially they are asking to solve $\displaystyle 0=-x^3+12x^2-37x+24$ for x.

I tried solving the equation using synthetic division. I knew that x must be < 3 since the smallest dimension is 3 feet to begin with. I also figured that x must be > 0 because it is not practical to remove a negative length from a dimension. So, the only two factors of 24 that satisfy these requirements are 1 and 2 (i knew zero was not an option since there is a constant in the polynomial equation). I tried using synthetic division for both 1 and 2 and both resulted in a remainder that did not equal zero. And of course, I know the answer is about .6, so its not a surprise that using whole number factors was not fruitful.

How can I figure out how much to remove from each dimension so that the resulting volume is three fifths of the original volume without using Cardano's method?

Prove It: I made an error, the problem in fact said that the sculptor wants the new volume to be three fifths OF the original volume.

Thank you to whoever is kind enough to read through this thread.