A rectangular column of cement is to have a volume of 20.25feet. The base is square, with sides 3 feet less than 1/2 the height of the column.

Question

What should the dimensions of the column be?

Answer 1

Let the height = x

Let the width and length  = (1/2)x - 3

 

The volume of a rectangle is

 

V = (width)²(height)

 

 

Plugging in the variables into the equation,

 

x[(1/2)x - 3)]² = 81/4

 

x[(1/4)x² - 3x + 9] = 81/4

 

(1/4)x³ - 3x² + 9x = 81/4

 

(1/4)x³ - 3x² + 9x - (81/4) = 0

 

 

Multiply both sides of the equation by 4.

 

x³ - 12x² + 36x - 81 = 0

 

Solve for x from this bolded equation.  But we need to factor this some more.  Use the rational root theorem as well as synthetic division to find a quadratic factor.

 

 

9 |   1      -12       36         -81

 

                  9      -27          81

___________________________

       1       -3          9           0

 

 

 

(x - 9)(x² - 3x + 9) = 0

 

 

We can accept x=9 as a solution to this equation.  This is your height.  Then plug in this value of x to find your width.