algebra
posted by lexis green .
Amy's project in metal shop class is to make a box (with no lid). Her shop teacher gave her a piece of metal that is 28 cm long and 20 cm wide. The assignment is to form the box by cutting squares out of each corner and bending the sides as shown below. The shop teacher, Mr. Rust, will give extra credit to any student who forms the box that holds the most when the cutout square is a whole number of centimeters on each side. Amy wants the extra point! How large a square should she cut from each corner to form the box with the greatest volume?

Let x = side of square to cut off from each corner
20  2x = width of bottom rectangle
28  2x = length of bottom rectangle
V = x(20  2x)(28  2x) = 4x(10  x)(14 x) =
V = 4x(140  24x + x²)
V = 4(140x  24x² + x³)
Differentiating and equating to zero
dV/dx = 4(140  48x + 3x²) = 0
3x²  48x + 140 = 0
x = {48 ± √[48²  4(3)(140)] } /2(3)
x = {48 ± √624] } /6
x = {48 ± 4√39] } /6
x = (24 ± 2√39) / 3
x = (24 + 2√39) / 3 = 12.16 Discard, > ½20
x = (24  2√39) / 3 = 3.8 inch
Use x = 4 since whole number of cm is required.
So each square to cut is 4 cm on the side.
4 cm squares gives a box of 4 x 12 x 20 = 960 cm^3
Respond to this Question
Similar Questions

calculus
A car[enter will be building a stained glass jewelry box. The frame of the box is to be built from a 48 inch stripe of metal which will be cut into 12 pieces. The 12 pieces are then soldered to form a rectangular box whose base is … 
Calculus
You are given a piece of sheet metal that is twice as long as it is wide an has an area of 800m^2. Find the dimensions of the rectangular box that would contain a max volume if it were constructed from this piece of metal by cutting … 
Calculas
A box lid is to be made from a rectangular piece of cardboard measuring 72 cm by 216 cm. Two equal squares of side x are to be removed from one end, and two equal rectangles are to be removed from the other end so that the tabs can … 
Algebra
A box is to be formed from a rectangular piece of sheet metal by cutting squares measuring 5 inches on a side and then folding the sides. The piece of sheet metal is twice as long as it is wide. If the volume of the box is to be 1760 … 
Calculus
A craftsman wants to make a cylindrical jewelry box that has volume, V, equal to 60 cubic inches. He will make the base and side of the box out of a metal that costs 50 cents per square inch. The lid of the box will be made from a … 
maths
a paper box without a lid is 25 cm long. 16 cm wide and 5 cm deep . how many square centimetres of paper have been used to make the box . 
math
The length, breadth and height of a closed box are respectively 2m, 1.5m and 0.8m. Find the total area of canvas required to cover itup. Find also the cost of canvas if 1sq m of canvas costs Rs. 30. A rectangular box without lid is … 
College Algebra
A rectanguler piece of metal is 5 inches longer than it is wide. Square with sides 1 inches longer are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 644 inches, what are … 
Physics
An airtight box has a removable lid of area 1.19E2 m2 and negligible weight. The box is taken up a mountain where the air pressure outside the box is 8.67E+4 Pa. The inside of the box is completely evacuated. What is the magnitude … 
physics
An airtight box has a removable lid of area 2.87 × 102 m2 and negligible weight. The box is taken up a mountain where the air pressure outside the box is 7.20 × 104 Pa. The inside of the box is completely evacuated. What is the …