The ratio of the heights of two similar rectangular prisms is 2:3. What is the ratio of their lateral areas?

- anonymous

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- LukeBlueFive

The area of two similar rectangles is equal to the square of their ratio.
Example: Rectangle A = 2x4, Rectangle B = 3x6. A's Area: 8, B's Area = 18. \[\frac{ 8 }{ 18 } = \frac{ 4 }{ 9 } = \left( \frac{ 2 }{ 3 } \right)^2\]

- anonymous

So what is the ratio of the lateral lines of two similar rectangular prisms with the ratio as 2:3?

- anonymous

8:27
2:5
4:9
1:0.33

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## More answers

- LukeBlueFive

The lateral lines of both of the similar rectangles will have a ratio of 4:9, which equals 2:3.
For example: Rectangle A = 1x5, Rectangle B = 2x10, Rectangle C = 3x15.
Also, the definition of similar rectangles states that two similar rectangles will have the same ratio for both their height and width.

- anonymous

Oh okay so the answer is 4:9?

- LukeBlueFive

Yes, sorry for answering the wrong question at first.

- anonymous

It's okay, I'm not good with math. All the numbers confuse me, wanna help me with another one? lol

- LukeBlueFive

lol, sure, why not?

- anonymous

Haha, okay as long as you explain how to do it I should be able to get it. I'm not asking you to do it for me :)
Part 1: Create and provide the dimensions for two similar figures of your choosing.
Part 2: What is the similarity ratio of these figures along with the ratio of their surface area and volume?
Part 3: Show your work, either using the actual volumes or using the formula, that the volume ratio is true.

- LukeBlueFive

Well, personally, I'd choose two small cubes and run the numbers on them, since that would make the math easier.

- anonymous

So what do I need to do? haha o.O

- LukeBlueFive

For the first part, give two pairs of three dimensions (one for each cube) that define the shape (such as 1x1x1). Since all cubes are similar, that part takes care of itself.

- LukeBlueFive

You with me so far?

- anonymous

|dw:1352233160153:dw|

- anonymous

like that? lol idk

- LukeBlueFive

Pretty much, except since the shapes need volume, they need to be 3D, not 2D.
|dw:1352233311609:dw|

- anonymous

Okay so is that part 1? How do I do part 2?

- anonymous

What are the dimensions on the 2 small cubes for part 1?

- LukeBlueFive

According to the question, it seems like you choose the dimensions. It seems to be teaching you how the equations and ratios work together.

- anonymous

Hm, I don't know how to do this! I'll have to wait and do it later, I have no idea what I'm supposed to do! haha

- LukeBlueFive

Maybe I'm not explaining it well. You just come up with two similar shapes, such as two cubes, give their dimensions, such as 1x1x1, 2x2x2, 3x3x3, etc., state what the ratio between them is (1:2, 1:3, etc.), calculate their surface areas (1, 4, 9, etc.) and volumes (1, 8, 27, etc.), then state those ratios are well.

- anonymous

I just don't know how to do it, I'm more of a visual learner. Let's see.. figure 1: 5 inches long, 2 inches wide, and 4 inches high. figure 2: 10 inches long, 4 inches wide, and 8 inches high. So figure 1 is half the size of figure 2 while figure 2 is 2x the size of figure 1? So the ratio of both figures would be 1:2? or 2:1?

- LukeBlueFive

Good example. The ratio depends on which figure you are comparing against the other, so both are correct. Figure 1 would have a ratio of 1:2 compared to figure 2, while figure 2 would have a ratio of 2:1 compared to figure 1.

- anonymous

I still need the area though, so is that l*w*h for each cube? if I'm right then figure 1: 40. and figure 2: 320? AHH this is so hard! lol

- anonymous

How do I find the volume of each cube?

- LukeBlueFive

Actually, you've already correctly calculated the volume of each figure.

- LukeBlueFive

Volume = l*w*h
Area = l*w

- anonymous

See now i'm lost -_-

- LukeBlueFive

Volume is for three dimensions, surface area is for two.
Your two figures will have three unique surface areas each (5x2, 2x4, and 5x4 for the first figure) since all three dimensions have different values.
|dw:1352234802496:dw|

- LukeBlueFive

Thankfully, all similar rectangles have the same ratios between all of their surface areas.

- LukeBlueFive

Just for reference, I think the ratio of the surface area of two similar shapes is equal to the square of their ratio and the ratio of the volume is equal to the cube of their ratio.

- anonymous

I'm just getting more confused.....

- LukeBlueFive

Okay, maybe it'll help if I give an example with units, to show the difference between calculating the surface area and the volume. It should also help explain how you can calculate the various ratios.

- LukeBlueFive

Let's say you have two cubes. Cube A is one inch on each side (height, width, and length), while Cube B is 2 inches on each side.

- LukeBlueFive

So the Cube A looks like this:
|dw:1352257542757:dw|
Cube B looks the same, just with 2's on each side.

- LukeBlueFive

To calculate the surface area, take any two edges and multiply them together. In this case, that would be: \[1 inch \times 1 inch = 1 inch^2\]

- LukeBlueFive

Volume is calculated similarly, but you multiply three sides together.
\[1 inch \times 1 inch \times 1 inch = 1 inch^3\]

- LukeBlueFive

Surface area should always have squared units, while volume should always have cubed units.

- LukeBlueFive

Now for Cube B's calculations,
Surface Area: \[2 inch \times 2 inch = 4 inch^2\]
Volume: \[2 inch \times 2 inch \times 2 inch = 8 inch^3\]

- LukeBlueFive

Now, moving on to ratios, you should just think of them as fractions. Cube A's sides are half the length of Cube B's, therefore, the ratio of Cube A's sides to Cube B's sides is 1:2. Cube A's surface area is a fourth that of Cube B's, so the ratio of Cube A's surface area to Cube B's surface area is 1:4. Cube A's volume is an eight of Cube B's volume, so the ratio of their volumes is 1:8.

- LukeBlueFive

As you can see, the ratio of their surface areas is equal to the square of the ratio of their sides. Put another way:
\[\frac{1}{4} = \left( \frac{1}{2} \right)^2\]
Also, the ratio of their volumes is equal to the cube of the ratio of their sides. Namely:
\[\frac{1}{8} = \left( \frac{1}{2} \right)^3\]

- LukeBlueFive

Does this make sense? If not, I'd be happy to explain specific points more clearly, if you can tell me what you don't understand.

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