Question

a rectangular box with a square base and open top is to be made from 1200cm^2 of material.

1. Find the dimensions of the largest box that can be made.
2.Use the second derivative test to show that the dimensions in 1 give a maximum volume.

Answer 1

So, The Total Surface of the box is 1200cm^2

Answer 2

the total of the box surface area is \[x ^{2}+4xh\] so do i equate
\[1200=x ^{2}+4xh\]

Answer 3

Let y^2 be the Area of the square base and h be the height of the box then,
y^2+4y*h=1200

Answer 4

h=(1200-y^2)/4y......i

Answer 5

volume is V=hbl then do i use the above equation to find h?

Answer 6

we seem to be thinking the same thing at the same time.

Answer 7

Now, VOLUME = AREA OF BASE * height

Answer 8

V=y^2 *h
V=y^2 *(1200-y^2)/4y

Answer 9

i see

Answer 10

V=y^2 *(1200-y^2)/4y
V=300y - 1/4 y^3

Answer 11

Now, Find the maximum value of V

Answer 12

how did you get V=300y - 1/4 y^3? i get x(1200-x^2)/4

Answer 13

That is same thing.

Answer 14

wait i see now my bad

Answer 15

only we are using different variable.

Answer 16

Find V'

Answer 17

so i find V'(x)=0 for maximum?

Answer 18

so \[300-\frac{ 3y ^{2} }{ 4 }=0\]

Answer 19

at x = 20 we will have a maximum?

Answer 20

YES

Answer 21

Now find height when x=20

Answer 22

so volume will be 4000cm^3

Answer 23

YEP

Answer 24

does y aply for H, B and L?

Answer 25

??

Answer 26

nvm. I have a height of 10cm and a width and breadth of 20cm?

Answer 27

YES

Answer 28

and the second part?

Answer 29

FIND v''

Answer 30

-3/2x

Answer 31

YES

Answer 32

Now, when x=20
V''=-3/2 *20 =-30

Answer 33

ok?

Answer 34

Which is negative, Thus V is maximum when x=20

Answer 35

so if the V'' is negative its maximum at point x and if it is positive then it is a minimum at point x?

Answer 36

is that just in general or is it for a fixed number?

Answer 37

U find x-coordinate of the stationary point FROM V'=0

Answer 38

Now, if for that x-coordinate , V'' is negative then the stationary point is maximum is positive then minimum and is 0 then neither maximum nor minimum.

Answer 39

thank you.

Answer 40

WElcome