Can you help me with my Precalculus?

Question

the question reads “a box is formed by cutting squares out of the corners of a flat piece of metal and folding the sides up. What size cuts should I make to a 14 inch by 20 inch piece to get a volume of 270 inches cubed?”

im so confused. Please help me out

Answer 1

If you draw the rectangle with squares cut off at the corners, your resulting dimensions will be

 

length = 20 - 2x

width = 14 - 2x

height = x

 

These are the dimensions of the box.  The height also represents the side length of the squares, since you fold the flat piece by the tabs to make the box.  Multiply these them together to get your volume.

 

V = x(14 - 2x)(20 - 2x)

V = x(280 - 68x + 4x²)

 

Set V=270.

 

270 = 4x³ - 68x² + 280x

 

 

Subtract 270 on both sides of the equation.

 

0 = 4x³ - 68x² + 280x - 270

 

 

You now have a cubic equation.  Use any method to solve for the zeros.  Rational root theorem, synthetic division, or ever factor by grouping.