Question

the volume of a cube is less than 25 , and the length of one of its edges is a positive integer. What is the largest possible value for the total area of the six faces?

Answer 1

Soo.....is it a cube? Cause all sides are equal then.

Answer 2

yeah its a cube

Answer 3

So all sides are equal then?

Answer 4

it would be the cubed root of the volume then

Answer 5

idk how to do that

Answer 6

What is the volume of a cube give that the height is equal to x?

Answer 7

i dont understand

Answer 8

well, it's a cube. What's the formula for the volume of a cube?

Answer 9

x*x*x=25 units^3
x^3=25 units^3
x=25^(1/3).......that gives you the length of the side.

Answer 10

volume = s3

Answer 11

right, the point here is that the volume of any 3-D rectangle is length x width x height. Since, by definition, a cube has equal length, width and height; we get that the volume is x * x * x. Now do we know what the maximum volume of the cube is?

Answer 12

less than 25

Answer 13

im losttttt

Answer 14

yeah i get that part

Answer 15

okay, so then we know that the MOST that the volume of our cube could EVER be is 25, yes?

Answer 16

yeah

Answer 17

so what if we let V=25 in our equation? If we solved for x that would give us the BIGGEST our height/length/width could EVER be, yes?

Answer 18

yup

Answer 19

okay, so then, what do we do next? What is our problem asking us to do?

Answer 20

for the largest possible value for the area of the six sides

Answer 21

and how would we go about doing that, given what we have solved for?

Answer 22

Any Ideas?

Answer 23

um im not suree

Answer 24

Any Ideas?

Answer 25

How do we find the area of a side of a cube?

Answer 26

add the sides up and divide?

Answer 27

Any Ideas?

Answer 28

In order to add up the sides, we need to know how to find a side, yes? How do we do that?

Answer 29

im not suree

Answer 30

Any Ideas?

Answer 31

what shape is the side of a cube?

Answer 32

square

Answer 33

Any Ideas?

Answer 34

and how do we find the area of a square?

Answer 35

we add the sides and divide

Answer 36

Any Ideas?

Answer 37

no, a square is flat, it has no sides.

Answer 38

multiply the base by itself

Answer 39

?

Answer 40

Any Ideas?

Answer 41

oh, silly software is janky. Yes we multiply the base by itself. So before we do that, let's solve for x. What do you get for x?

Answer 42

im lost

Answer 43

What was our volume equation?

Answer 44

V=x^3

Answer 45

and we said our max volume was?

Answer 46

25

Answer 47

so, let's solve for x

Answer 48

1

Answer 49

nope, try again. Remember, we're saying: 25 = x^3 and solving for x.

Answer 50

25^1/3

Answer 51

exactly, and what does that equal?

Answer 52

the side

Answer 53

well, I meant numerically?

Answer 54

2.924017738

Answer 55

?

Answer 56

hold on, lemme break out my calculator..

Answer 57

k

Answer 58

yep, that is correct, but let's just say it's 2.924 for simplicity. so what does that x mean?

Answer 59

2.924

Answer 60

?

Answer 61

in terms of the cube, what does that x value represent?

Answer 62

the side

Answer 63

?

Answer 64

no, it doesn't represent a side. Remember, we said that the formula for the volume of a cube is what?

Answer 65

V=x^3

Answer 66

exactly, and what do those "x"s represent? Hint: Is it height, length, width or all of them?

Answer 67

all of them

Answer 68

?

Answer 69

right

Answer 70

exactly right. And we can't say that length, height, or width are a side of the cube. Because a side of the cube is equal to length * width. It's VERY important to remember the difference between a line, area, and volume.

Answer 71

ohh ok

Answer 72

but back to our problem, so now that we know that, what can we say the sides of the maximized cube? What is a single side equal to?

Answer 73

what can we say about the sides*

Answer 74

brb

Answer 75

25^1/3?

Answer 76

?

Answer 77

remember what we said earlier, that value is only the length/width/height of the cube. We want a single side of the cube, which is 2-D or a square. so how do we find the area of a square/side of the cube?

Answer 78

25^1/3 x 2

Answer 79

?

Answer 80

Haha, excellent! Let's look back at the problem what is it asking us to find again?

Answer 81

the largest possible value for the area of 6 sides

Answer 82

?

Answer 83

So, the problem wants 6 sides, so let's multiply our value for one of the sides by 6. (Recall that the height/width/length is 2.924, and that we said the area of a side is (2.924) x 2)

Answer 84

2.924x2x6?

Answer 85

right-o, so what does that equal?

Answer 86

35

Answer 87

right

Answer 88

?

Answer 89

correct, and does this satisfy our problem? Is that the MAXIMUM area of the 6 sides?

Answer 90

i dont think soo

Answer 91

?

Answer 92

Indeed it does! Remember that the volume we used was the MAXIMUM volume the cube could ever be. Which means that the length/height/width was the maximum length/height/width of the cube and thus we solved for the maximum area for a side of the cube. Does that make sense?

Answer 93

ohhh okay so its 35

Answer 94

it should indeed be 35.