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anonymous

  • one year ago

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

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  1. mathstudent55
    • one year ago
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    When you change the side of a solid by a scale factor of k, the volume changes by a factor of \(k^3\)

  2. mathstudent55
    • one year ago
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    Your scale factor is 1/2 What is \((1/2)^3\) ?

  3. anonymous
    • one year ago
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    idk

  4. mathstudent55
    • one year ago
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    We need to find this: \((\dfrac{1}{2}) ^3\) Have you learned exponents?

  5. mathstudent55
    • one year ago
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    For example, what is \(3^2\) ?

  6. anonymous
    • one year ago
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    18

  7. anonymous
    • one year ago
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    \[1\frac{ 1 }{ 2 }\]

  8. mathstudent55
    • one year ago
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    No, this is how exponents work. \(3^2 = 3 \times 3 = 9\) For example, \(4^3 = 4 \times 4 \times 4 = 64\)

  9. mathstudent55
    • one year ago
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    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 \(3^2 = 3 \times 3 = 9\)

  10. mathstudent55
    • one year ago
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    Let's use a higher exponent. What is 2^4? You need to multiply four 2's together. \(2^4 = 2 \times 2 \times 2 \times 2 = 16\)

  11. anonymous
    • one year ago
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    ok then what

  12. mathstudent55
    • one year ago
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    Do you have a better understanding of exponents now?

  13. anonymous
    • one year ago
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    yes

  14. mathstudent55
    • one year ago
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    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.

  15. anonymous
    • one year ago
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    \[1\frac{ 1 }{ 2 }\]

  16. mathstudent55
    • one year ago
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    According to what the exponents mean, \(\left( \dfrac{1}{2} \right)^3 = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} \)

  17. anonymous
    • one year ago
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    \[1\frac{ 1 }{ 2 }\]

  18. mathstudent55
    • one year ago
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    \(1 \dfrac{1}{2} = \dfrac{1}{2} + \dfrac{1}{2} +\dfrac{1}{2} \) We are not adding the fractions. We are multiplying them together.

  19. mathstudent55
    • one year ago
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    \(\left( \dfrac{1}{2} \right)^3 = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{2 \times 2 \times 2} = \dfrac{1}{8}\)

  20. anonymous
    • one year ago
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    o ok

  21. mathstudent55
    • one year ago
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    1/8 is quite different from 1 1/2

  22. mathstudent55
    • one year ago
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    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.

  23. anonymous
    • one year ago
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    ummm dont get it

  24. mathstudent55
    • one year ago
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    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 ?

  25. anonymous
    • one year ago
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    30

  26. mathstudent55
    • one year ago
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    Correct.

  27. mathstudent55
    • one year ago
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    Here is an explanation with a figure that sometimes makes things easier to understand.

  28. anonymous
    • one year ago
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    so 30 is the ans

  29. mathstudent55
    • one year ago
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    |dw:1435895261732:dw|

  30. mathstudent55
    • one year ago
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    Yes, the answer is 30 cm^3

  31. anonymous
    • one year ago
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    thank u i have like 3 more can u help me with them

  32. mathstudent55
    • one year ago
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    In the figure above the cube has a side of 4 cm. The volume of the cube is \(V = s^3 = (4 ~cm)^3 = 4 ~cm \times 4 ~cm \times 4 ~cm = 64 ~cm^3\)

  33. mathstudent55
    • one year ago
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    Now let's draw a smaller cube in which every side is half of the original one. |dw:1435895446522:dw|

  34. mathstudent55
    • one year ago
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    The volume of the new cube is: \(V = s^3 = (2 ~cm)^3 = 2 ~cm \times 2 ~cm \times 2 ~cm = 8 ~cm^3\)

  35. mathstudent55
    • one year ago
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    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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