Question

An open metal tank with a square base is made from 12m^2 of sheet metal. Find the length of the side of the base for the volume of the tank to be a maximum and find this maximum volume

Answer 1

i think you should use inequalities, and finaly draw a graph using them. Right sstarica?

Answer 2

hmmm, it's more like an optimization problem where you have to find the Absolute maximum in this case.

Answer 3

he base is a square, but the other parts take shape of a cylinder, weird ._.

Answer 4

the*

Answer 5

I can't figure out a sketch for this one >_<, with no sketch, no formula, no answer

sorry

Answer 6

maxima problem.. determine what subject to be maximize or minimize; in this case, the volume of a tank..
V = lwh
and in this case the l is equal to the w. therefore
V = w^2*h
then, give the constraint of the problem.. in this case; the material to be use.. only 12 m^2
Make an equation for the constraint: that is, the formula for the surface area..

Answer 7

you mean L not I right?

Answer 8

yeah, L..

Answer 9

so, now we have the general function which is:

V = w^2h

Answer 10

the constraint: the surface area of a tank, cube with open on the top, which means there are 5 faces only. one on the bottom, and four on the sides.
The surface area on the bottom is w^2
and the surface area on the side is wh. since there are 4 side faces, 4w^2
therefore the total surface area is:
w^2 + 4wh = 12 m^2(meter squared)

Answer 11

from the general function stated by sstarica, express h in terms of w from my total surface area equation,
then we will have a function of V in terms of w..

Answer 12

Or an identity.

Answer 13

The problem states only the base is made from material that is 12 sq material.

Answer 14

Never mind it does probably include the sides in the that 12 sq meters of material. Come on fellows go ahead and work this out. I can't envision construction of this tank without some waste in the material (cutting out the corners). I guess they could just press it into a mold!!!

Answer 15

Can we safely assume that the base is a square? i.e. length = width.

Answer 16

as what i understand the problem. the tank is square base, and the tank is made from 12m^2. The problem does not define how the tank is constructed.

Answer 17

use JBWeld :)

Answer 18

do we assume that the sheet metal is a square to begin with? or simply 12m^2 of rectangular shape?

Answer 19

most of these problems have you cut corners and fold the sides..... but that is an assumption for this one....

Answer 20

for me, the assumption is that the total surface area is 12 m^2. :)

Answer 21

Cant we just buy a tank thats already large enough to use? :)

Answer 22

lol., then you are the consumer, not the designer.

Answer 23

I say we go to the ag store and buy the tank.

Answer 24

If we assume only the base is 12 sq meters then we have no limit on the height of the sides that means you could have a mighty big tank!

Answer 25

yeah, that is why it must be assumed that the total surface area of the tank is 12 m^2.
I think I already give the equation for that..
using that assumption we can have a solution.

Answer 26

whats the measurements of the sheet metal tho? is it:
1,12
2,6
3,4
or some decimalized monstrosity?

Answer 27

like 4sqrt(3), 4sqrt(3) ?

Answer 28

uhh..that shoulda been 2sqrt(3) s but you know....the keyboard hates me :)

Answer 29

also, it just says tha the base is square; it doesnt say that the tank itself is square.... hmmmm

Answer 30

I tried to solved it. the answer is:
width = 2m
length = 2 m (for square base)
height = 1 m
therefore the total volume is 4 m^3,
the total surface area is 12 m^2.
^_^

Answer 31

That works out for the amount of material.....but is it the maximum volume, I'm thinking it is but can't prove it.

Answer 32

try use the equation:
V = w^2 * h (square base)
Surface area = 12 m^2 = w^2 (square base) + 4 * w * h (four faces of the sides of the tank)
solve for h in terms of w:
h = (12 - w^2) / 4w
therefore, the volume becomes
V = (w^2) * (12-w^2)/4w
or
V = (12w - w^3)/4
then derive V with respect to w

Answer 33

That would be w=2 I guess that is the proof that is needed. Thanks for the step by step procedure.

Answer 34

\[V=(12w-w ^{3})/4=3w-w ^{3}/4=3w-1w ^{3}/4\]

Answer 35

\[dV/dw=3-3w ^{2}/4\]

Answer 36

wow., how did you do that radar?

Answer 37

Set to 0
\[3-3w ^{2}/4=0\]
\[3w ^{2}/4=3\]
Multiplying both sides by 4/3
\[w ^{2}=4\]

Answer 38

Hey, Janjor did the ground work! lol

Answer 39

yeah!
i think i don't understand what you mean by "ground"
lol

Answer 40

Set up the foundation (foundations are usually on the ground)