Question

The shipping company BigBlue has a size requirement for rectangular packages: the sum of the three dimensions cannot exceed 135 inches. If one end of the package is a square with sides of x inches, and the other dimension is y inches, what are the dimensions that will maximize the volume?

Answer 1

Is this calculus? My main question is can I solve it using calculus?

Answer 2

yes, i think the first derivative too. but if you cant just tell me what you know

Answer 3

I can. I think I got it. Are you ready?

Answer 4

yayy

Answer 5

we know that 2x + y = 135

Answer 6

Vmax = (x^2)y

Answer 7

Lets get the volume equation in terms of x

Answer 8

so we solve the first and substitute. Vmax=(x^2)(135-2x)

Answer 9

y=135-2x?

Answer 10

lets distribute x^2... Vmax=135x^2-2x^3

Answer 11

Are you with me so far? or is there is step that confuses you?

Answer 12

x = 45 and 0

Answer 13

no go on

Answer 14

not yet. I think we must take the derivative next and find x when the slope = 0

Answer 15

1st derivative is 270x-6(
x^2)

Answer 16

so Vmax' =270x-6x^2

Answer 17

nice.

Answer 18

Set Vmax' = 0 and solve for x

Answer 19

yep x= 45 and 0

Answer 20

ok. haha nice. I didn't know you were that far already.

Answer 21

i just needed to go past the substitution, thank you... Gosh ill love you forever

Answer 22

i guess x=0 is extraneous

Answer 23

Your welcome. I guess the final anser would be 45 x 45 x 45

Answer 24

y= 135-2(45)

Answer 25

I wouldn't have guessed a perfect cube in the beginning but I think the math is all right.

Answer 26

me too

Answer 27

how do i find the The maximal volume ?

Answer 28

45*45*45?

Answer 29

Well we just found the x value where the volume would be at its maximum. So now just use V=lwh. So 45^3

Answer 30

can i throw in another question?

Answer 31

its alittle tricky and i got stuck?

Answer 32

A crate open at the top has vertical sides, a square bottom, and a volume of 864 cubic meters. If the crate has the least possible surface area, find its dimensions.
Hint : Each surface of the box is what shape?

Answer 33

first thing we need to take into account is there is no top

Answer 34

V=864cm^3
V=hx^2
SA= x^2 + 4xh

Answer 35

Those are the equations I came up with based on the information given

Answer 36

i always have a problem figuring out how to do the substitution

Answer 37

ok. So solve the Volume equation for h and tell me what you get

Answer 38

so i guess h = 864cm^2/x^2

Answer 39

sorry cm^3

Answer 40

ya so that IS h. so rewrite the SA equation and where there is an h put what it is in terms of x

Answer 41

do the powers subtract? since eventually x^2 is cm^2 ?

Answer 42

leave off the units if it is confusing you for now

Answer 43

SA(x) = x^2+ 3456x/x^2

Answer 44

ok. Do you see anywhere where that equation can be simplified

Answer 45

2x-3456x^-1 ?

Answer 46

S = x^2 + 3456x^-1
S' = 2x - 3456x^-2
S' = 2x - 3456/x^2

Answer 47

3456=2x^3 ?

Answer 48

Thats what I did

Answer 49

x=12

Answer 50

Plug it into our original equations and I think we have our answer

Answer 51

y=6

Answer 52

Correcttttt!!!!

Answer 53

I have to head to class. Good luck

Answer 54

thanks

Answer 55

good luck to you too