anonymous
  • anonymous
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
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
http://static.k12.com/calms_media/media/395500_396000/395703/2/df976dddee0bcc6657252f77ac772d47dd775716/53252_question.jpg
mathstudent55
  • 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
  • mathstudent55
Your scale factor is 1/2 What is \((1/2)^3\) ?

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anonymous
  • anonymous
idk
mathstudent55
  • mathstudent55
We need to find this: \((\dfrac{1}{2}) ^3\) Have you learned exponents?
mathstudent55
  • mathstudent55
For example, what is \(3^2\) ?
anonymous
  • anonymous
18
anonymous
  • anonymous
\[1\frac{ 1 }{ 2 }\]
mathstudent55
  • mathstudent55
No, this is how exponents work. \(3^2 = 3 \times 3 = 9\) For example, \(4^3 = 4 \times 4 \times 4 = 64\)
mathstudent55
  • 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
  • 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
  • anonymous
ok then what
mathstudent55
  • mathstudent55
Do you have a better understanding of exponents now?
anonymous
  • anonymous
yes
mathstudent55
  • 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
  • anonymous
\[1\frac{ 1 }{ 2 }\]
mathstudent55
  • 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
  • anonymous
\[1\frac{ 1 }{ 2 }\]
mathstudent55
  • 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
  • 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
  • anonymous
o ok
mathstudent55
  • mathstudent55
1/8 is quite different from 1 1/2
mathstudent55
  • 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
  • anonymous
ummm dont get it
mathstudent55
  • 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
  • anonymous
30
mathstudent55
  • mathstudent55
Correct.
mathstudent55
  • mathstudent55
Here is an explanation with a figure that sometimes makes things easier to understand.
anonymous
  • anonymous
so 30 is the ans
mathstudent55
  • mathstudent55
|dw:1435895261732:dw|
mathstudent55
  • mathstudent55
Yes, the answer is 30 cm^3
anonymous
  • anonymous
thank u i have like 3 more can u help me with them
mathstudent55
  • 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
  • mathstudent55
Now let's draw a smaller cube in which every side is half of the original one. |dw:1435895446522:dw|
mathstudent55
  • 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
  • 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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