Question

http://prntscr.com/b6pe8l
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Someone want to help me with this design. I'm a bit confused about what to do.

Answer 1

I chose a rectangular prism for my cereal box design.

Answer 2

I already wrote down the SA and V.

Answer 3

It's the rest I'm having trouble with.

Answer 4

In packaging design, you would probably want to choose a shape that gives you the biggest bang for you buck. In other words, you would want to pack as much of your product into as little packaging material as possible.

Answer 5

Can you tell me which part of which assignment you are stuck at?

Answer 6

How would you calculate the volume and surface area of a standard cereal box at home?

Mainly the questions from the second link. : )

Answer 7

Do you have a box of cereal and a ruler or tape measure?

Answer 8

Yes I do!

Answer 9

Though, do I measure the width, length, and height in inches?

Answer 10

Well then, what dimensions of the box would you need to measure in order to calculate Surface Area and Volume?

Answer 11

Yep, those would be the 3 dimensions for which you need a measurement.

Answer 12

You said you already have SA and V written down, do you mean you already have the formulas for a rectangular prism?

Answer 13

Yes. I'm measuring right now. Brb.

Answer 14

A quick google search gave me this: "Cereal box size depends on the cereal brand and the volume size. For example, a box that is 12 inches long, 7 5/8 inches wide and 2 1/2 inches deep is required to hold 12.8 ounces of Multi Grain Cheerios. It varies according to the cereal."

Answer 15

Alright, I got W=8ft , H=12 ft, and L=3ft

Do I times each one by twelve to convert to inches now?

Answer 16

@mrm

Decided to go by feet.

Answer 17

You have a cereal box that is 12 feet high?...

Answer 18

It looks like you measured in inches... why are you writing it in feet?

Answer 19

Haha. Oops.

Alright, what do we do now?

Find the volume by lxwxh and then square it?

Answer 20

You need to find the surface area of the box. To find the surface area of the box, you need to add up the area of all the faces of the box.

Answer 21

So, it's 2(8x3)+2(3x12)+2(8x12)

Answer 22

Yep. Then calculate the cost of the box.

Answer 23

How do we calculate the cost of the box?

Answer 24

@openstudier013087

Answer 25

What did you get for your surface area?

Answer 26

312

Answer 27

@openstudier013087

Answer 28

Then I'm guessing you do 312x0.05

Answer 29

@zepdrix

Help me finish this <3?

Answer 30

@Kikuo make sure we are on the same page; where are designing a new cereal box?

Answer 31

Yep. I currently found the V and SA of the cereal box.

Calculate the cost of manufacturing a standard cereal box if cardboard costs $0.05 per square inch.

Answer 32

you have 312" for your surface area?

Answer 33

Yep!

Answer 34

312*0.05

Answer 35

need to clarify volume and surface area?
what volume measure did you get?
surface area is 312"

Answer 36

I got 288

Answer 37

your volume is 288
your surface area is 312
manufacturer cost is 15.60$

Answer 38

the new design need to have the same volume but with a small surface area, so we can start my calculating surface area of other shapes that have the same volume.

Answer 39

Wait is it 312x0.05

Answer 40

312*0.05 = 15.60$ = manufacturer cost

Answer 41

Aright how do we do the rest?

Answer 42

@latinC

Answer 43

Why not try a cylinder?


If the volume is fixed to be 288 cubic inches, then V = 288


Let's say the height is 12 inches (1 ft). So h = 12.


Volume of a cylinder


V = pi*r^2*h
288 = pi*r^2*12


solve for r and tell me what you get.

Answer 44

I solve for r for both V and 288?

Answer 45

you replace V with 288

Answer 46

Ah alright

4.8989794855663562

@jim_thompson5910

Answer 47

I don't know how you got that. It's not the correct value of r.

Answer 48

I divided 288 by 12 and took the square root of that

Answer 49

you forgot about the pi

Answer 50

You have to divide both sides by \(\Large 12\pi\) then take the square root of both sides.

Answer 51

How did you get 12pi?

@jim_thompson5910

Answer 52

From the formula
V = pi*r^2*h

Answer 53

I made h = 12

Answer 54

V = 288
h = 12


V = pi*r^2*h
288 = pi*r^2*12
288 = 12pi*r^2

Answer 55

Ah, I see what you did there. Alright.
2.7639531957706838311

This is what I got.

Answer 56

yes

Answer 57

Now use

r = 2.7639531957706838311
h = 12

to find the surface area of this cylinder


Formula
\[\Large SA = 2\pi*r^2+2\pi*r*h\]
SA = surface area of cylinder
r = radius
h = height

Answer 58

Question, why are finding the SA instead of the volume?

Answer 59

@latinC said
`your volume is 288`
`your surface area is 312`
`manufacturer cost is 15.60$`
which is in reference to your rectangular prism (aka rectangular box)


So the goal is to find some other 3D solid where it has the same volume (288) but has a smaller surface area.


Why smaller surface area? Because the smaller the surface area, the less cardboard needed. The less cardboard, the less the total cost per package.

Answer 60

So we took the volume of V = 288 and we let the height be 12 inches.

You solved for r to get r = 2.7639531957706838311

Now use this to find the surface area, ie, find out how much cardboard you'll need to buy.

Answer 61

Alright! Makes sense.

I got 256.2921905381722457

Answer 62

How much would it cost if it costs $0.05 (5 cents) per square inch?

Answer 63

12.814609526908612285

Answer 64

so approx $12.81

Answer 65

@latinC said
`manufacturer cost is 15.60$`
which is in reference to your rectangular prism (aka rectangular box)


The cylinder costs $12.81 to wrap up. So the cylinder package is cheaper by about $2.79 (15.60-12.81 = 2.79)

Answer 66

Thanks so much! Would you help checking some answers of mine and helping with two other problems?

Answer 67

The rectangular box has a volume of 288 cubic inches. It costs $15.60 for all the cardboard.


The cylinder has a volume of 288 cubic inches. It costs $12.81 for all the cardboard.


So clearly the cylinder is the better choice. You get the same volume but at a smaller cost. Win win if you ask me.


Why haven't cereal boxes been turned to cylinders? A few reasons

1) People may be used to boxes and prefer boxes over anything else. If it were different, people might not buy them

2) Boxes are easier to stack and pack in tightly. There is no empty space when you put boxes together. With cylinders, you have unused space

3) Boxes may be easier to manufactor and put together. They may hold up easier too

4) If the cereal was in a cylinder, then it may look like a big pringles can, which is probably what cereal companies want to avoid.

I'm sure there are other reasons too. These reasons are just educated guesses though.

Answer 68

I agree.

Answer 69

It turns out that if you want the biggest bang for your buck, and you want to minimize the volume as much as possible, then go for a sphere.


https://en.wikipedia.org/wiki/Sphere


Quote from that link

`Of all the solids having a given volume, the sphere is the one with the smallest surface area`


So if the company wanted to save the most money, they'd go for a sphere. Again there are many reasons why this is a bad idea. One of them is that the spheres would roll off the shelf. Another reason is that there would be empty unused space when you pack them in boxes.