## 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

1. anonymous
2. mathstudent55

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

3. mathstudent55

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

4. anonymous

idk

5. mathstudent55

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

6. mathstudent55

For example, what is $$3^2$$ ?

7. anonymous

18

8. anonymous

$1\frac{ 1 }{ 2 }$

9. mathstudent55

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

10. 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 $$3^2 = 3 \times 3 = 9$$

11. mathstudent55

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

12. anonymous

ok then what

13. mathstudent55

Do you have a better understanding of exponents now?

14. anonymous

yes

15. 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.

16. anonymous

$1\frac{ 1 }{ 2 }$

17. mathstudent55

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

18. anonymous

$1\frac{ 1 }{ 2 }$

19. 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.

20. mathstudent55

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

21. anonymous

o ok

22. mathstudent55

1/8 is quite different from 1 1/2

23. 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.

24. anonymous

ummm dont get it

25. 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 ?

26. anonymous

30

27. mathstudent55

Correct.

28. mathstudent55

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

29. anonymous

so 30 is the ans

30. mathstudent55

|dw:1435895261732:dw|

31. mathstudent55

Yes, the answer is 30 cm^3

32. anonymous

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

33. mathstudent55

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

34. mathstudent55

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

35. mathstudent55

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

36. 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.