if a sheet of metal is a square of side 24 cm from each corner equal squares are removed and formed into an?

Question

open box find the maximum volume

Answer 1

Let x be the sides of the small squares. Then the volume V of the box is W*L*H where W=24-2x, L=24-2x and H=x. Thus,
V = WLH = (24-2x)^2*x
V = 4x^3 - 96x^2 + 576x
Now, differentiate once and equate to 0 and solve for x
dV/dx = 12x^2-192x+576 = 0
x = 12 or x = 4
Now insert these values of x in V and determine the maximum volume
V(12) = 4*12^3 - 96*12^2 + 576*12
V(12) = 0
V(4) = 4*4^3 - 96*4^2 + 576*4
V(4) = 1024 cm^3
then the side of the squares should be 4 cm to get the maximum box volume of 1024 cm^3.

Answer 2

512 cm3
To get 6 (a cube has 6 sides) equal boxes you have to cut the sheet into a grid of nine squares.
Why?
If you were to cut a aquare into 6 equal pieces, you'd wind up with shapes 8x12 cm. So to get 6 8x8 cm squares to make up your cube, you need to cut as if on a 3x3 grid.
Then it's simply l*w*h and boom, cubic volume.

Answer 3

because of fact that we are reducing out squares of length x from each and each corner, we choose 2 xs to have the means to in high quality condition in the two course, so x is between 0 and 20 cm. the quantity then will become (50-2x)(40-2x)x. increasing that provides 4x^3 - 180x^2 + 2000x. Differentiating: 12x^2 - 360x + 2000 = 0 Or 3x^2 - 90x + 500 = 0. Use the quadratic formulation to discover x = 22.637 or x = 7.362. because of fact that x is below 20 cm, x would desire to be 7.362 cm, to 3dp.

Answer 4

let the side of the square removed from the sheet be x cm
then the volume of the box will be given by (25-2x)(25-2x)*x
and dV/dx=d/dx(4x^2-100x+625)*x= [x(8x-100)+4x^2-100x+625]
for maximum this must be zero. Equating
12x^2-200x+625=0=>12x^2-150x-50x+625=0=>(6x-25) (2x-25)=0
so x=25/6 ignoring 25/2 as inadmissible
now maxima check d2V/dx^2=24x-200 and when x=25/6
24*25/6-200=-100 check answered
so the side of the squares cut from the corners=25/6cm

Answer 5

I presume that this just a light hearted brain teaser as your box will be 8 x 8 x 8 which means that the internal volume will be 8 cubed which is 512 cubic (what was it? oh yes French measure)

Answer 6

Have you got maths tomorrow and only just remembered your homework?

I'm 22 I don't do homework

Answer 7

good work meckaban. Thats perfect....