Question

A rectangular box is to be made to the following requirements:
- The length must be one and a half times the width
- The twelve edges must have a total length of 6m

Find the dimensions of the box that meets these requirements and that maximizes the volume.

Answer 1

Please help, using differentiation

Answer 2

your width=w=0.4meters = 0.4m
height h=0.5m
length L=1.5w=1.5x.4=0.6
volume v=Lwh=0.6(4)(.5)=3/25= 0.12meter cube

Answer 3

do you have the answer of choices there fetzer?

Answer 4

Those are exactly correct, would you be able to explain as to how you obtained those answers :). Thank you very much

Answer 5

ok bear with me fetzer..lol what ststes are you in? ill explain them very clearly...

Answer 6

Ststes? do you mean States? I'm in Western Australia.

Answer 7

ah ok,,,,

Answer 8

draw there a rectangle in your scratch paper

Answer 9

Yeah, don't wait for me just keep going :p

Answer 10

ok sorry man..lol..

Answer 11

you have width=w
length = 1.5w
perimeter P=4w+6w+4h
P=10w +4h but it is given as 6meters
P=10w +4h =6meter
4h =6 - 10w
h=(3-5w)/2
now the volume V=Lwh
V=1.5w(w)(h)
V= 1.5w^2(h)

Answer 12

now plug in your h=(3-5w)/2 to the volume V= 1.5w^2(h)

Answer 13

Okay I am still with you :)

Answer 14

V= 1.5w^2[(3-5w)/2]
V=[(9/2)w^2 -(15/2w^3)]/2
V=[(9/2)w^2 -(15/2)w^3)]/2
V=[9w^2 -15w^3]/4

Answer 15

factored out 3/4 there

Answer 16

now you can do derivative of V with respect to w

Answer 17

now solve for w

Answer 18

V=(3/4)(3w^2-5w^3)
dV/dw =(3/4)(6w-15w^2)=0 we zero them to get the maximum volume.

15w^2=6w
15w = 6
w=6/15=0.4

Answer 19

Now L=1.5w=1.5(.4)=0.6m
h=(3-5w)/2 =(3-5(.4))/2 =0.5
V= Lwh=.6(.5)(.4)=3/25=0.12m

Answer 20

im sorry V=0.12meters cube

Answer 21

Thank you very much

Answer 22

yw fetzer..good luck now....

Answer 23

fetzerc,

A solution using Mathematica is presented in the attachment. Unfortunately a solution as presented in the attachment cannot be produced in a couple of minutes.

Answer 24

How do you use Mathematica

Answer 25

Mathematica 8 is a very large (3Gb installation file) and comprehensive mathematics computer program able to generate symbolic as well as nummeric solutions. The program is not free, but the price for active students is very low. More about Mathematica can be found at:

http://www.wolfram.com/solutions/education/students/

As a math hobbyist I own a Home edition of Mathematica 8 and have no financial interests in the Wolfram organization.

Answer 26

A friend actually has a student copy and I'm wanting to useit, could you give a tutorial on how I would use it for the question I asked :P?

Answer 27

Yes I guess I could help out. Mathematica is a very large system and the help file(s) is about 1Gb in size. The student copy should have a help system as part of it. I will have to admit that the thing is very intimidating at first.