I'm lost in these math skills questions....please help...or figure it out!?

Question

1.Calculate the ratio of surface area to volume for a cube that is 2 cm high. What happens to this ratio as the size of the cube decreases?

2.If a 4-cm cube doubled just one of its dimensions?length, width, or height?what would happen to the ratio of surface area to volume?

....
This is what you know:
A cube has 6 equal sides of 4 cm X 4 cm.
This is what you want to find:
the ratio ( R) of surface area to volume for each cube
These are the equations you use:
surface area ( A) = width X length X 6
volume ( V) = length X width X height
R = ( A)/( V)
Solve for surface area and volume, then solve for the ratio:
A = 4 cm X 4 cm X 6 = 96 cm?
V = 4 cm X 4 cm X 4 cm = 64 cm?
R = 96 cm?/64 cm? = 1.5 cm?/cm?
Check your answer by multiplying the ratio by the volume. Do you calculate the surface area?

Answer 1

1.Calculate the ratio of surface area to volume for a cube that is 2 cm high. What happens to this ratio as the size of the cube decreases?

Ratio = surface area / volume
surface area = 6 * length ^2 (6 sides, each side's area is l^2)
volume = length ^3 (l * w * h where l = w = h).

ratio = 6l^2 / l^3
l^2 / l^3 becomes 1/l, so:
ratio = 6/l

At l = 2, the ratio = 6/2 = 3/1.
As the length decreases, the ratio increases. At l = 1, the ratio becomes 6/1.

2.) If a 4-cm cube doubled just one of its dimensions?length, width, or height?what would happen to the ratio of surface area to volume?

The surface area of a box is 2 lw + 2wh + 2lh.
The volume = l * w * h.
Let l = w, and h be different for the above question. This leaves:
sa = 2 l^2 + 2lh + 2lh = 2l^2 + 4lh
v = l^2 * h
ratio = (2l^2 + 4lh) / (l^2 * h)

Split the fraction:
2l^2 / (l^2 * h) + 4lh / (l^2 * h)
ratio = 2/h + 4/l

When l and h = 4 cm, the ratio = 2/4 + 4/4 = 1.5.
If h increases to 6 cm, the ratio = 2/4 + 4/6 = 7/6 = 1.16666.

Thus, as h increases, the ratio of surface area to volume decreases.

Answer 2

Don't worry about the numbers yet. Think about a cube of any size.
Surface area of cube = W x L x 6 (sides)
But since a cube has W = L = H, then
Surface area = 6 x W x W

(a)
Volume of a cube = W x L x H = W x W x W
6So the ratio of Surface Area to Volume = (6 x W x W) / (W x W x W)
or simply: 6 / W, for whatever dimension of a cube.
In your case, since W = 4, then 6/4 = 1.5
As the size of the cube decreases, the ratio 1.5 remains constant

(b) If the size of the cube is changed in just one dimension, then the area formula is different. There are 2 sides with area = L x W plus 4 sides with area = W x H: (2 x L x W) + (4 x W x H). Let's say that L = W, but H is different. Then the formula is (2 x W x W)+(4 x W x H). Therefore, with L & W = 4 cm, but H = 8 cm, the new surface area = (2x4x4)+(4x4x8) = 32+128 = 160 sq cm.
The volume is simply L x W x H = 4 x 4 x 8 = 128.

So the new ratio becomes 160 / 128 = 1.25

Hope this helps.

Answer 3

Your formulas are correct. Use them to solve both problems.

1) S.A. = 2 cm x 2 cm X 6 = 24 cm^2
Vol = 2 cm x 2 cm X 2 cm = 8 cm^3
R = A / V = 24 cm^2 / 8 cm^3 = 3 cm^-1

Let's say the cube decrease to 1 cm per length.
S.A. = 1 cm x 1 cm X 6 = 6 cm^2
Vol = 1 cm x 1 cm X 1 cm = 1 cm^3
R = A / V = 6 cm^2 / 1 cm^3 = 6 cm^-1

So as the size of the cube decreases, the ratio of Surface Area to Volume increases.

2). You have the first part of #2 in your question.
For the second part, think of each face of the cube.
You will have 4 faces that are 8 cm x 4 cm and then the two end faces that are still 4 cm x 4 cm. (Only one direction is going to increase. You may want to draw a cube)
A = (8 cm X 4 cm X 4 faces) + (4 cm X 4 cm X 2 faces ) = 160 cm^2
V = 8 cm X 4 cm X 4 cm = 128 cm^3
R = 160 cm^2 / 128 cm^3 = 1.25 cm^-1

Therefore, if you only increase one side, the ratio of surface area to volume decreases.

Answer 4

First off that I would point out that R is a ratio and is best described as "x to y". 1.5 actually represents the ratio 3/2 or 3 to 2.
the ratio of Area to Volume of a 2cm cube would be 3/1 (24/8) or 3 to 1.

In the last part, it is important that you realize that it is no longer a cube and the surface area formula must reflect that. Four sides remain the same, but two sides increase to 4X8. therefore the formula is area= (4X4X4 + 2X4X8) or 64+64 = 128. The volume then changes to 4X4X8 which equals 128 as well. the ratio of Area to Volume then becomes 1/1 or 1 to 1.

Ratios are often listed as fractions or percentages rather than a decimal. (ex. a road grade of 6% or a stair run of 11:6... 11 tread and a six inch rise).

Answer 5

I'm not good in math, but I hope this site helps you.


http://www.mathforum.org/dr.math/