This prism has a volume of 240 cm3.
What would the volume of the prism be if each dimension was halved? (Scale factor ishttp://static.k12.com/bank_packages/files/media/mathml_68f74329bc2e147090cb3e6f27fbce4255c06f3b_1.gif .)
A.
120 cm3
B.
480 cm3
C.
30 cm3
D.
60 cm3

- anonymous

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- schrodinger

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- anonymous

http://static.k12.com/calms_media/media/395500_396000/395703/2/df976dddee0bcc6657252f77ac772d47dd775716/53252_question.jpg

- mathstudent55

When you change the side of a solid by a scale factor of k,
the volume changes by a factor of \(k^3\)

- mathstudent55

Your scale factor is 1/2
What is \((1/2)^3\) ?

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## More answers

- anonymous

idk

- mathstudent55

We need to find this: \((\dfrac{1}{2}) ^3\)
Have you learned exponents?

- mathstudent55

For example, what is \(3^2\) ?

- anonymous

18

- anonymous

\[1\frac{ 1 }{ 2 }\]

- mathstudent55

No, this is how exponents work.
\(3^2 = 3 \times 3 = 9\)
For example,
\(4^3 = 4 \times 4 \times 4 = 64\)

- mathstudent55

The base tells you which number is going to be multiplied.
The exponent tells you how many of the bases to use.
\(3^2\)
The base is 3, so 3 will be multiplied.
The exponent is 2, so there will be two bases multiplied, or two 3's multiplied.
That means
\(3^2 = 3 \times 3\)
and
\(3 \times 3 = 9\)
so \(3^2 = 3 \times 3 = 9\)

- mathstudent55

Let's use a higher exponent.
What is 2^4?
You need to multiply four 2's together.
\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)

- anonymous

ok then what

- mathstudent55

Do you have a better understanding of exponents now?

- anonymous

yes

- mathstudent55

Ok.
In your problem, the side was changes by a scale factor of 1/2
The rule is if the sides changes by a factor of k, the volume changes by a factor of k^3.
In your case, that means the volume changes by a factor of
\(\left( \dfrac{1}{2} \right)^3\)
Now we need to use our knowledge of exponents and find what
\(\left( \dfrac{1}{2} \right)^3\) is equal to.

- anonymous

\[1\frac{ 1 }{ 2 }\]

- mathstudent55

According to what the exponents mean,
\(\left( \dfrac{1}{2} \right)^3 = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} \)

- anonymous

\[1\frac{ 1 }{ 2 }\]

- mathstudent55

\(1 \dfrac{1}{2} =
\dfrac{1}{2} + \dfrac{1}{2} +\dfrac{1}{2} \)
We are not adding the fractions. We are multiplying them together.

- mathstudent55

\(\left( \dfrac{1}{2} \right)^3 = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{2 \times 2 \times 2} = \dfrac{1}{8}\)

- anonymous

o ok

- mathstudent55

1/8 is quite different from 1 1/2

- mathstudent55

Ok. Now we know that the volume factor is 1/8 when the side factor is 1/2.
That means that when the side of a solid becomes half what it used to be, the volume becomes 1/8 of what it used to be.

- anonymous

ummm dont get it

- mathstudent55

If your prism started with a volume of 240 cm^3, and each dimension became half, that means the volume is only 1/8 of 240 cm^3.
What is 240 cm^3 divided by 8 ?

- anonymous

30

- mathstudent55

Correct.

- mathstudent55

Here is an explanation with a figure that sometimes makes things easier to understand.

- anonymous

so 30 is the ans

- mathstudent55

|dw:1435895261732:dw|

- mathstudent55

Yes, the answer is 30 cm^3

- anonymous

thank u i have like 3 more can u help me with them

- mathstudent55

In the figure above the cube has a side of 4 cm.
The volume of the cube is
\(V = s^3 = (4 ~cm)^3 = 4 ~cm \times 4 ~cm \times 4 ~cm = 64 ~cm^3\)

- mathstudent55

Now let's draw a smaller cube in which every side is half of the original one.
|dw:1435895446522:dw|

- mathstudent55

The volume of the new cube is:
\(V = s^3 = (2 ~cm)^3 = 2 ~cm \times 2 ~cm \times 2 ~cm = 8 ~cm^3\)

- mathstudent55

Since the new cube has a side that is half of the original cube (2 cm is half of 4 cm), the scale factor of the sides is 1/2.
That means the scale factors of the volumes is (1/2)^3 = 1/8
Now look at the volumes: 64 cm^3 and 8 cm^3
Sure enough, 8 cm^3 is 1/8 the volume of the original cube, 64 cm^3.

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