Question

An open box constructed from a tin sheet has a square base and a volume of 216 in^3. Find the dimensions of the box, assumng that the minimum amount f material was used in its construction

Answer 1

i have no clue where to go

Answer 2

this is optimization btw

Answer 3

put x = length of square base. then the base has area
\[x^2\]

Answer 4

the height is h but the volume is \[x^2\times h = 216\] solving for h gives \[h=\frac{216}{h}\]

Answer 5

now the surface area is \[x^2 + 4xh \] yes?

Answer 6

area of base plus 4 times area of sides

Answer 7

oh damn i made a typo.
solving for h gives
\[h=\frac{216}{x^2}\]

Answer 8

no the nonsense i wrote.

Answer 9

sorry i slipped away from the computer

Answer 10

so all in all we get a surface area of
\[A(x)=x^2+4\times x \times \frac{216}{x^2}\]
\[A(x)=x^2+\frac{54}{x}\]

Answer 11

thats ok i made a stupid typo

Answer 12

surface area is same as volume right...

Answer 13

oh heck no

Answer 14

volume is length times width times hight. 3 dimensions

Answer 15

sorry can you hold on for a sec while i try to catch up to what your doing please

Answer 16

surface area is an area. 2 dimensions. i wait. ignore the line where i wrote
\[h=\frac{216}{h}\] it should be
\[h=\frac{216}{x^2}\]

Answer 17

oh ok lol sorry i just saw it the sites really laggy

Answer 18

true that

Answer 19

how'd u get the surface area?

Answer 20

main goal is obviously to find a function that gives the surface area. then we minimize that function

Answer 21

ok surface area

Answer 22

base is a square with side x, so area is
\[x^2\]

Answer 23

4 sides have base x and height h so surface area of each side is
\[xh\]

Answer 24

there are 4 of them so we have all together \[x^2+4xh\]

Answer 25

oh ok

Answer 26

area of base plus 4 times area of sides. but we also have a "constraint' namely that the volume is 216 cubic whatevers

Answer 27

inches lol

Answer 28

so we can solve for h in terms of x via
\[V= 216 = x^2\times h
\]

Answer 29

making
\[h=\frac{216}{x^2}\]

Answer 30

oh ok and thats how u got the x^2 + 4*x*216/x^2

Answer 31

now we plug that back in to the formula for the total surface area to get a function of one variable namely

\[A(x)=x^2+\frac{54}{x}\]

Answer 32

exactly

Answer 33

:) lol it just hit me... sorry

Answer 34

oh wait you have it and i made a mistake

Answer 35

yeah the numerator right? of the 2nd term

Answer 36

\[\frac{4\times 216\times x }{x^2}=\frac{864}{x}\]

Answer 37

i divided instead of multiplied, brain fart

Answer 38

so now we have to find the derivative

Answer 39

lol

Answer 40

so finally, assuming i did not make another bone head mistake, we have
\[A(x)=x^2+\frac{864}{x}\]

Answer 41

2x-864x^-2

Answer 42

derivative is not to much work here.
\[\frac{d}{dx} A(x)=x^2-\frac{864}{x^2}\]

Answer 43

2x yes

Answer 44

\[\frac{d}{dx}A(x)=2x -\frac{864}{x^2}\]

Answer 45

set this beast = 0 and solve

Answer 46

i got the same thing i just could hav took my x back to the bottom

Answer 47

almost do it in your head. numerator is \[2x^3-864\]

Answer 48

there is a critical point at 0 but we can ignore it because it makes no sense to have a box with base 0

Answer 49

yay! im doing the same thing

Answer 50

\[2x^3-864=0\]
\[x^3=432\]
\[x=\sqrt[3]{463}\]

Answer 51

whatever that is

Answer 52

7.736 rounded

Answer 53

how's we do?

Answer 54

7.56 its 434

Answer 55

yeah well it is late

Answer 56

half of 864 =432 so we are both wrong!

Answer 57

7356 rounded

Answer 58

7.56 i mean

Answer 59

hmm yes i have it on my scraps and im typing the wrong thing

Answer 60

you got the right answer in any case. good work!

Answer 61

thanks you too as always but do i find my minimum by plugging in x=0, x = 7.56
what are my endpoints?