Question

Geometry question.. will medal!!

The theme park company is building a scale model of the killer whale stadium main show tank for an investor's presentation. Each dimension will be made 6 times smaller to accommodate the mock-up in the presentation room. How many times smaller than the actual volume is the volume of the mock-up?

Answer 1

When using scale factors, higher dimensional quantities (such as area or volume) are always effected in the same fashion. It can be shown that this is true in general, but I'll show to specific cases:

Rectangle. Area of a rectangle is L*W
If we multiply all sides by a number, a, we get new lengths:
L' and W', and a new area: A' = L' * W'
Since L' = aL and W' = aW, the area becomes:
A' = aL*aW = (a^2)*L*W

Looking at a rectangular block, Volume = L*W*H (length, width, height)
Multiplying each side by a factor a:
L' = aL, W' = aW, H' = aH
The new volume is:
V' = aL*aW*aH = (a^3)L*W*H

As we can see, the area gets effected by the number a, but squared. The volume is multiplied by the number a, but cubed. This is true in general. We can see this with a circle/sphere, as well:

A= pi*r^2
Radius gets multiplied by a: r' = ar
New Area:
A' = pi*(ar)^2 = (a^2)*pi*r^2

V= (4/3)*pi*r^3
r'=ar

V' = (4/3)*pi*(ar)^3 = (a^3)*(4/3)*pi*r^3

Once again, the area gets an a^2 and the volume gets an a^3.

a is called the "scaling factor." It's the number that we are multiplying each side by. What is your a?

Answer 2

Thank you! So what equation should I be using? The main show tank has a radius of 70 feet and the mock-up should be 6 times smaller.

Answer 3

So, if you want to take something and make it 6 times smaller, what would you do? (What number would you multiply by/divide by?)

Answer 4

The new dimensions of the tank will just be divided by 6, correct? 70/6 = 11.66667 And then would I have to find the volume of each shape or is there a formula that I can use instead?

Answer 5

Divide by 6, right. This is the same as multiplying by 1/6. This is your "a."

Now, you could go and calculate the new volume of each shape and take a ratio, but we aren't given the original dimensions.

However, we know what happens to a volume when you multiply each of it's physical dimensions by a common factor.

Answer 6

The dimensions are given earlier in the lesson, actually. The radius is 70 feet long of the original tank. Sorry for not mentioning that earlier :P I'll post a pic in a second.

Answer 7

\[V_{new} = a^3*V_{old}\]

What we want to know is:
\[\frac{V_{new}}{V_{old}} = ?\]

Answer 8

We're dealing with the main show tank, which is on the left.

Answer 9

Wait, wouldn't it be volume of old divided by the volume of new? Since the volume of the original shape will be larger.

Answer 10

You can actually just use the first equation, if you know the original volume. We want to know the new volume.

Answer 11

Your question, as stated, is "How many times smaller than the actual volume is the volume of the mock-up?"

So we'd need to find:
\[\frac{V_{new}}{V_{old}}\]

Answer 12

So tell me if I'm on the right track with this... 70/6 is 11.66667 so I need to find the volumes of both shapes. The volume of the original shape with a radius of 70 is 359007 cubic feet. If the volume of the new shape is approx. 1662, then I would need to divide 359007/1662 to find my answer? Or should I divide 1662/359007?

Answer 13

1662/359007

Answer 14

You can get a more exact answer, though.

Answer 15

So it would be 0.00462943619 times smaller? How can I get a more exact answer? Thanks for helping.

Answer 16

Well, it's be 0.00462943619 times as large, yes. We have to be careful about terminology.

Remember when we said something was 6 times smaller, we divided by 6 (or multiplied by 1/6)? We want to know how much smaller it is, so the answer is actually 1/0.00462943619, which should be around 200 or so.

To get a more exact answer, we use:
\[V_{new} = a^3 * V_{old}\]

When we want to know how much SMALLER something is, what we are asking is:
\[V_{new} = \frac{1}{C}*V_{old}\]
Find C.

Comparing these two equations, we can see:
\[\frac{1}{C} = a^3\]
Rearranging:
\[C=\frac{1}{a^3}\]

Since we are multiplying each dimension of the tank by 1/6, a = 1/6.
Therefore:
\[C = \frac{1}{a^3} = \frac{1}{(\frac{1}{6})^3} = 6^3 = 216\]

So, the model is 216 times smaller than the original.

Answer 17

Ah that makes sense, thanks. But wouldn't I get the same answer if I just divided the old volume by the new? 359007/1662 = 216 which is the same answer.

Answer 18

You do, but I wanted to make sure you understood the terminology. Perhaps I was being too nit-picky. I wanted to make sure you were noting the "How many times SMALLER" part :)

Answer 19

Okay thank you :) you were very helpful! Do you mind if I ask a quick question related to this problem? I already have the answer but I'm not 100% sure it's correct.

Answer 20

sure

Answer 21

Using the same information from the previous problem, what percent of change occurred from the actual tank to the mock-up of the tank?

Here is my answer:

To find the percent of change, you would need to divide the change over the original value. In this case, to find the percent of change that occurred from the actual tank to the mock-up of the tank, we need to first find the amount of change between the two volumes by subtracting.
359007 - 1662.069 = 357344.931
Now that we know the change, we need to divide that by the original volume which in this case is the volume of the actual killer whale main show tank.
357344.931/359007 = 0.9953703716
Next, we need to multiply that by 100 to obtain a percent.
0.9953703716 x 100 = 99.53703716
The percent of change that occurred from the actual tank to the mock-up of the tank is approximately a 99.54% decrease.

Answer 22

Do you think 99.54% decrease sounds about right?

Answer 23

It's right :)

\[PercentChange = \frac{V_{old}-V_{new}}{V_{old}} = \frac{V - \frac{1}{216}V}{V} = 1-\frac{1}{216} = \frac{215}{216}\]

Which is approximately .99537 or 99.54%

Answer 24

I'd specify that you are talking about the volume, to be sure, though.

Answer 25

"The percent change in the volume is..."

Answer 26

Thank you so much! You're a lifesaver :)

Answer 27

My pleasure :)