## anonymous 5 years ago 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?

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

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.

3. myininaya

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

4. myininaya

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

5. myininaya

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

6. myininaya

we have like terms

7. myininaya

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

8. myininaya

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

9. myininaya

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

10. myininaya

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

11. anonymous

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

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

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

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