Question

Caculus help please! A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions b=8in by a=10in by cutting out equal squares of side x at each corner and then folding up the sides how much do we cut from each corner in order to maximize the volume?

Answer 1

Let x = the dimension of the corner cut. This is also the height.

Answer 2

let lenght =10-2x
width = 8-2x
Volume =l x w x h
V = x (8-2x)(10-2x)
Complete and differentiae set to 0 and solve. got to run good luck.

Answer 3

ok so V'(x)=1*(8-2x)(10-2x)+x(-2)(10-2x)+x(8-2x)(-2)

Answer 4

we can clean that up so finding x from V'=0 is easier

Answer 5

V'(x)=80-16x-20x+4x^2-20x+4x^2-16x+4x^2

Answer 6

we have like terms

Answer 7

V'(x)=8x^2-72x+80 unless i made a mistake

Answer 8

V'(x)=8(x^2-9x+10)

Answer 9

i did make a mistake V'=12x^2-72x+80

Answer 10

now set V'=0 to find critical numbers let me know if you have anymore trouble

Answer 11

this helps a lot but i can't factor 12x^2-72x+80=0. and to find x: the cut out corners are x by x

Answer 12

Divide all terms by 4 getting \[3x ^{2}-18x+20=0\]

Answer 13

a=3, b=-18, and c\[(18\pm \sqrt{84})/6\]

Answer 14

\[(18\pm2\sqrt{21})/6\] Results in x approx 1.4724 in.