Word Problem

Question

A 7in thick slice is cut off the top of a cube, resulting in a rectangular box that has volume 49in^3. Find the length of the original cube. Round answer to two decimal places.

Answer 1

Hi Angela.

 

A cube has all four sides the same length.  

 

The volume of a box is length x width x height, but a cube has length = width = height.  So we will call the this "x".

 

Now the volume of the cube is (x) (x) (x) = x³

 

If you cut 7 inches off of the top, your height is no longer "x", but "x-7".

 

Now the volume of our box (it is no longer a cube) is length x width x height = (x) (x) (x - 7) = 49

 

We need to find "x" and we will know the length ( and width and height) of the original cube.

 

We need to solve (x) (x) (x-7) = 49.  First rewrite it as:

 

x²(x-7) = 49

 

Distribute

 

x³ - 7x² = 49

 

Move everything to the left:

 

x³ - 7x² - 49 = 0

 

and find the roots of this polynomial.

 

I put this on a polynomial solver and got  x = 7.80447 inches

 

 

Answer 2

Suppose the original cube has a length of s. Then the length and width of the cropped off cube is also s and its height is s - 7. The volume is then s²(s-7). Set this equal to 49 and solve for s:

 

s²(s-7) = s³ - 7s² = 49

 

s³ - 7s² - 49 = 0.

 

In order to solve this you can try guessing a solution or graph the left side on a calculator and locate a root.

Guessing is too hard in this case, so if you graph the left side and use a zero finder, you can find the approximate solution, which is about 7.80 inches.