Functions Question

Question

identical squares are cut from each corner of a rectangular sheet of tin 8cm x 6cm. the sides are bent upward to form an open box. if the volume of the box is 16cm^3, what is the length of each side of the squares cut from the original sheet?

Answer 1

Call the length of the small square you call out "d".  Then when you cut the squares and fold up the sides, the box will have dimensions: (8-2d) x (6-2d) x d

 

The volume of this is: d ( 8-2d) (6-2d) = d * 2(4-d) * 2(3-d) = 4d(4-d)(3-d) = 16 (given)

Then d(4-d)(3-d) = 4 = d(12-7d+d²) = 4

 

Multiplying everything out gives the following cubic:

 

d³ - 7d² + 12d -4 = 0

 

Try simple integer factors of 4: 1, 2 & 4.  2 satisfies the equation.

 

divide the cubic by (d-2) using synthetic division will then give you that it can be written as:

 

(d-2) (d² - 5d +2)

 

Using the quadratic formula on the quadratic in the parentheses gives that the other two solutions are:

 

d = 0.5*(5+/- sqrt(25-4*2)) = 0.5 *(5+/- sqrt(17)) = 4.56 and 0.43845

 

4.56 cannot be another length of the side since twice that is greater than the lengths of the original sheet.

 

However, 0.43845 (this is equal to 0.5*( 5-sqrt(17)) will work.

 

So the two solutions are d = 2 and d=0.43845