SOLUTION: An open box is to be made from a rectangular sheet of tin by cutting out equal squares from each corner and folding up the sides. If the sheet is 21 cm long and 16 cm wide. Express

Algebra ->  Volume -> SOLUTION: An open box is to be made from a rectangular sheet of tin by cutting out equal squares from each corner and folding up the sides. If the sheet is 21 cm long and 16 cm wide. Express      Log On


   

Question 754288: An open box is to be made from a rectangular sheet of tin by cutting out equal squares from each corner and folding up the sides. If the sheet is 21 cm long and 16 cm wide. Express the volume V of the box as a function of x where x is the length of the square to be cut from each corner of the sheet.
a) If the volume of the box made is 5816 cm^3 find x.
Can you please help me out? Thanks so much in advance:)
Can you also please show the steps it would really help me understand:)
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Using the non-upturned part as the base area, this area is 21-2x in one direction and 16-2x in the other direction. The base area is then %2821-2x%29%2816-2x%29 square units.

The box is x unit tall. The volume is then %2821-2x%29%2816-2x%29x cubic units.
You will use this in its unfactored form, so perform the multiplications.
%28336-32x-42x%2B4x%5E2%29x
4x%5E3-42x%5E2-32x%5E2%2B336x
4x%5E3-74x%5E2%2B336x

You were given a volume value of 5816 cubic centimeters. You want to know what is x.
4x%5E3-74x%5E2%2B336x=5816
Divide b.s. by 2
highlight%282x%5E3-37x%5E2%2B168x-2908=0%29

Do you know about testing for upper and lower bounds? If you do not, still no big worry, because you already know that your starting sheet is a short as 16 cm. You will NOT need any root to test that is greater than this 16. Also note, you are interested in REAL roots; not complex roots with imaginary numbers.

Factoring 2908 in preparation to use Rational Root Theorem:
2%2A2%2A727
Yes, That's it! ________ 727 is a prime number; just try and factor it.

The roots to check according to Rational Roots theorem and using synthetic division will be positive 1, 2, and 4. One of them will be found to give a zero remainder and therefore be one of the roots of the cubic equation highlighted, and will then be the value of x to give the given volume value.