Question

OPTIMIZATION: According to U.S. postal regulations, the girth plus the length of a parcel sent by mail may not exceed 108 inches, where by "girth" we mean the perimeter of the smallest end. What is the largest possible volume of a rectangular parcel with a square end that can be sent by mail? Such a package is shown below. Assume y>xy>x. What are the dimensions of the package of largest volume?

Find a formula for the volume of the parcel in terms of x and y.
Volume =?

Answer 1

Volume will be something like \(\large V = x^2y\) if x is the dimension of the square base.

The length plus girth would be \(\large 108 = 4x+y\), since the perimeter (girth) of the square end is 4x.

Solve the second eq. for y, plug it into the first. Differentiate, set equal to zero, solve.

You might want to use a first or second derivative test to check it's a maximum.

Then plug the x value back into the volume equation, to work out the maximum volume.

Answer 2

need help @Amenah8

Answer 3

I'm trying to do it based on what @agent0smith said but I'm still a bit confused

Answer 4

well almost all of my people are offline

Answer 5

i just messaged someone

Answer 6

i would call @zepdrix but he is offline

Answer 7

@Jamierox4ev3r help

Answer 8

do you know someone smart in your fan

Answer 9

yeah i messaged them hopefully they come XD

Answer 10

but i'm doing ok with solving so far

Answer 11

@osprey

Answer 12

oh you here @osprey

Answer 13

how did you know she needed help @zepdrix

Answer 14

ug i got the derivative to be -12(x-18)x, set it equal to 0 and solved for x which I got to be 0

Answer 15

is that right?

Answer 16

this is another optimisation problem. I'm in danger of getting out my linear programming and other minimising/maximising stuff ... Unf I don't have it to hand (nor to brain).
@bonnieisflash1.0 should I say Miaow ? ("Batman returns" is a "nice" film, and "Miss Kitty" steals the show completely.

Answer 17

go in your nest and hide there @osprey

Answer 18

@bonnieisflash1.0 ouch ! will do

Answer 19

okay

Answer 20

-12(x-18)x for derivative? whuuu?
Where is 18 coming from??

Answer 21

I was trying to do what agentsmith said: V=x^2 * y, and length plus girth = 108=4x+y

i solved for y using the second equation and got y=108-4x and then plugged that into the Volume equation and took the derivative

Answer 22

So I was trying to take the derivative of V=X^2 * (108-4x)

Answer 23

So product rule, yes?

Answer 24

right

Answer 25

(x^2)(-4) + (108-4x)(2x)

Answer 26

ok good

Answer 27

2(108-4x)x-4x^2 ??

Answer 28

Put the x in front, it's ugly back there >.<
2x(108-4x)-4x^2

Ok sooooo you've got your derivative of volume.
Look for the critical value which will maximize volume.

Answer 29

wait don't i have to solve for x?

Answer 30

@zepdrix

Answer 31

Yes, that's what it means to find critical points, solve for x.