I don't understand how to do this math problem...?

Question

1) Aaliyah needs to figure out how many 4-inch cubes will fit in the large box. When the large box is closed, the top & bottom are each 192 sq. inches. The front and back are each 128 sq. inches. The sides are each 96 sq. inches.

Questions to answer:
~What are the dimensions of the large box?
~How many 4-in. cubes will fit in the large box?

2)
Aaliyah creates the same product, bu the dimensions are cm. instead of in. How does the volume of a 4 cm cube compare to the volume of a 4 inch cube? (Actually, I understand this one. But you can do it if you want to._

Thanks, but this stuff still confuses me. I'll ask my friend to help me tomorrow. =D I'm bad at geometry-type stuff.... not when I have numbers though.

Answer 1

1) You need to solve these equations first...

W*L = 192
L*H = 128
H*W = 96

Let's multiply the first two equations by each other:
W*L * L*H = 192 * 128
H*W * L^2 = 24576

Now divide this by the last equation:
L^2 = 24576 / 96
L^2 = 256
L = 16.

Now you can figure the rest of the dimensions:
W*L = 192
W*16 = 192
W = 192/16
W = 12

W*H = 96
(12)*H = 96
H = 96/12
H = 8

So the dimensions are:
8 inches high, 12 inches wide, 16 inches long.

That's 2 cubes high (2 x 4" = 8"), 3 cubes wide (3 x 4" = 12"), 4 cubes long (4 x 4" = 16").

All together that is 24 cubes (2 x 3 x 4 = 24)

2) If you shrink everything by a factor of 2.54 (to get centimeters) the answer doesn't change at all. Since every dimension changes to cm, the box shrinks and so do the cubes.

Your dimensions will be 8 cm x 12 cm x 16 cm. With a block that is 4 cm on each side, you still fit 2 blocks x 3 blocks x 4 blocks = 24 blocks.

Answer 2

This is a question of volume, V. For a cube, V= l x w x h (length x width x height).

If you remember that area, A, is length x width and given in square units (inches in this case), then V= A x h.

You are given your A: 192 sq inches -- which is the bottom, or base, of your box.

The trick here is finding the height.

If the front/back is 128 sq. inches, then you know the AREA of the front/back (watch your units!). The sides are 96 sq inches. You have to make equations and solve for h.

Drawing a picture may help you.

Here's what you know:
l x w = 192 (top/bottom)
l x h = 128 (front/back)
w x h = 96 (side)

That means you have 3 terms. You have to rearrange so you can solve for one term at a time (let's do l first):

Solve one for h: h= 128/l
then substitute into another: w x (12/l) = 96
Solve another for w: w= 192/l
then substitute into the other: (192/l) x (12/l) = 96
Now you only have one term!
2304 / l^2 = 96
2304 = (96)(l^2)
24 = l^2
12 = l (remember your units were square inches at first. Now it's just inches)
So, if length = 12 inches, we can solve for h:
h = 128/l = 128 / 12

Use that to find V: V= 192 x (128/12) = 2048 inches CUBED

But this only solves part of your problem. You are filling the box with 4 inch cubes. The volume above is how many 1 inch cubes would be needed to fill the box. Divide that number by 4, and you'll have your final answer:

2048 / 4 = 512 4-inch cubes will fill the box.

Answer 3

of give the box's sides variable names
lets call the width (from left to right) x
the depth (from front to back) y
and the height (from the bottom to the top) z

the top and bottom surface area would then be x*y
the front and back surface area would then be x*z
and the left and right surface area would then be y*z

as far as the cm part goes, the total volume would be 1/(2.54)^3 smaller or .06102... of the size of the inch box, but would still hold 24 cm cubes
since:
xy=192
xz=128
yz=96

we can solve for the dimensions of the box x,y, and z
manipulate yz=96
z=96/y

xz=128
substitute from above
x*96/y=128
manipulate
x=y*128/96

xy=192
substitute from above
y*y*128/96=192
y^2=144
or y=12

x=192/y=192/12
x=16

yz=96
so z=96/y=96/12
z=8

so you have a box thats 12x16x8

since it is 12 inches wide, you could stack 3 blocks wide
since it is 16 inches deep, you could stack 4 block deep
and since it is 8 inches tall, you could stack 2 blocks high
giving a total of 3*4*2=24 blocks that will fit inside.

Answer 4

1) Area = length * width
Set the three sides to variables called a, b, and c, and write the formulas for the areas:

a * b = 192
b * c = 128
a * c = 96

Solve for one and substitute into another:

b = 192/a
c = 96/a
192/a * 96/a = 128
(192 * 96)/a² = 128
a²/(192 * 96) = 1/128
a² = (192 * 96)/128
a = 12
b = 192/a = 129/12 = 16
c = 96/a = 96/12 = 8

So, you have your dimentions (12 x 16 x 8), and now you have to figure out how many 4-in cubes will fit: That's 2 (across) * 3 (up) * 4 (deep) = 24 4-inch cubes. :-)

2) There are 16.387064 cm³ in an in³ (that's 2.54³).