Question

Two similar prisms have heights 4 cm and 10 cm. What is the ratio of their surface areas?

A) 2:5
B) 4:25
C) 4:10
D) 8:125

Thanks!

Answer 1

What does it mean if they're 'similar' ?

Answer 2

Would this be A?

Answer 3

Their heights reduced?

Answer 4

Well that's a good place to start. However, what does it mean that they are 'similar' ?

Answer 5

It means that they are comparable...

Answer 6

Specifically it means that the ratio of their dimensions will be the same right?

Answer 7

Okay

Answer 8

aha so c?

Answer 9

If the heights have a ratio of 2:5, the dimensions of their bases will also have a ratio of 2:5.

Answer 10

I'm lost... obviously

Answer 11

Ok look. You know that the base of the prisim has some dimensions.. maybe it's a square, maybe it's a rectangle. Whatever it is, the dimensions (width and height) will have a 2:5 ratio with the base of the larger prism.

Answer 12

So if the area of the base is \(L \times W\) then the ratio of the area will be :
\[2L \times 2W \over 5L \times 5W\]

So it's going to have a 4:25 ratio in area.

Answer 13

Because it has a 2:5 ratio in dimension.

Answer 14

okay, so just times the dimensions by themselves? Or was that just with this problem? Thank you!

Answer 15

If you want I can try for a more detailed explanation that might make it easier to understand.

Answer 16

a

Answer 17

That was my initial guess, but polak worked it out and said 4.25?

Answer 18

Ok, look (since this will be helpful in the next problem):

Let \(W_1\) be the width of the smaller prisms base.
Let \(L_1\) be the length of the smaller prisms base.
Let \(H_1\) be the height of the smaller prism.

Let \(W_2\) be the width of the larger prisms base.
Let \(L_2\) be the length of the larger prisms base.
Let \(H_2\) be the height of the larger prism.

So we have that the two are similar and that:
\[H_1 = \frac{2}{5}H_2\]
Therefore (because they are similar)
\[L_1 = \frac{2}{5}L_2\]\[W_1 = \frac{2}{5}W_2\]

Therefore the area of the bases will be
\[A_1 = C(L_1 \times W_1)\]
Where C is some constant pertaining to the shape of the base (1/2 for triangles, etc)

Therefore:
\[A_2 = C(L_2\times W_2)\]
And their ratios are \[\frac{A_1}{A_2} = {C(L_1 \times W_1) \over C(L_2\times W_2)}\]\[= {L_1 \times W_1 \over L_2\times W_2}\]\[={\frac{2}{5}L_2 \times \frac{2}{5}W_2 \over L_2\times W_2}\]\[=\frac{2}{5}\times\frac{2}{5} \times {L_2 \times W_2 \over L_2\times W_2} = {4 \over 25}\]

Answer 19

Thanks so much for helping me! I wish I could understand all this though, lol :(

Answer 20

Which part is confusing you?

Answer 21

I have trouble comprehending just how exactly I would do those steps on paper, or with my calculator...

Answer 22

No calculator required. You'd do it on paper just the way I did. The question is, do you understand what I did?

Answer 23

If not, which part doesn't make sense?

Answer 24

We can go through it more slowly.