anonymous one year ago Aran moved a cone-shaped pile of sand that had a height of 6 ft and a radius of 3 ft. He used all of the sand to fill a cylindrical pit with a radius of 6 ft. How high did the sand reach in the pit? Use 3.14 to approximate pi and express your final answer in tenths.

1. mathstudent55

You need the formulas for the volumes of a cone and a cylinder.

2. anonymous

3.14xr^2xh/3

3. anonymous

and 3.14xr^2xh

4. mathstudent55

Correct. Here they are written in LaTeX which makes it easier to understand: $$\Large V_{cone} = \dfrac{1}{3} \pi r^2 h$$ $$\Large V_{cylinder} = \pi r^2 h$$

5. mathstudent55

Now compare the formulas. If you have a cone and a cylinder, and they both have the same radius of the base and the same height, how do the volumes compare?

6. anonymous

We don't know the height of the cylinder...Right/ I really don't understand

7. mathstudent55

Let me explain better what I was asking above. I'm not talking about your problem now. This is just in general. Let's say you have a cone and a cylinder. The cone and the cylinder have the same radius. The cone and the cylinder have the same height. What will be the difference in their volumes? To answer this question, just look at the formulas for the volumes. What is the only difference you notice between the volume formulas?

8. anonymous

1/3?

9. mathstudent55

|dw:1432764194813:dw|

10. mathstudent55

Exactly. The only difference is the 1/3 in the cone's volume formula. That means for the same height and radius of a cone and a cylinder, the cone's volume is 1/3 the size of the cylinder's volume.

11. mathstudent55

Ok. Now let's deal with your problem.

12. mathstudent55

Since you know the formula for the volume of a cone, you can use it to find the volume of the original pile of sand, which is a cone with given radius and height.

13. anonymous

169.65?

14. mathstudent55

|dw:1432764569511:dw|

15. mathstudent55

Great. Now you need the formula for the volume of a cylinder. Write the formula with the info you know, r = 6, leave the height as h, and set it equal to the volume of the cone you just found above. Then solve for h.

16. anonymous

3.14*6^2*h 3.14*36*h 113.04*h

17. anonymous

169.65=113.04*h

18. anonymous

169.65-113.04=56.61=h?

19. anonymous

Nope it's wrong ummm I don't know...

20. mathstudent55

Wait. Just one question. In finding the volume of the cone, did you divide by 3 like the formula requires?

21. mathstudent55

|dw:1432765558766:dw|

22. anonymous

wait... That's the answer? that doesn't make sense to me at all.. I got 49.2

23. mathstudent55

No. The drawing above just shows the volume of the cone. I think when you calculated the volume of the cone, you forgot to divide by 3. The volume of the cone is 56.52 ft^3, not 169.56 ft^3

24. anonymous

oh.. hehe sorry, my mistake...

25. anonymous

so 56.52=113.04xh?

26. anonymous

So h=0.5?

27. anonymous

Actually no.. I'm just confusing myself more and more

28. mathstudent55

Ok. Now for the volume of the cylinder: |dw:1432766341984:dw|

29. mathstudent55

Yes, your cylinder volume is correct. Now we equate the two volumes: |dw:1432766436217:dw|

30. mathstudent55

Your equation is correct. Now you need to solve for h. Since h is being multiplied by 113.04, you need to do the opposite operation, that is, you need to divide both sides by 113.04

31. mathstudent55

|dw:1432766554551:dw|

32. mathstudent55

h = 0.5 ft You are correct. Great job!

33. anonymous

Oh my gosh! Thank you so much.....

34. anonymous

Can you check this answer though?

35. anonymous

Remi has two cone-shaped containers with the same diameter. He will place the smaller container inside the larger one. Before he does this, he wants to fill the larger container with water so that it will be completely full but won’t spill when he places the smaller container inside. The diameter of both containers is 12 cm. The height of the smaller container is 6 cm, and the height of the larger container is 18 cm. What volume of water must Remi put in the large container? Use 3.14 to approximate pi and express your final answer in hundredths. 452.16

36. mathstudent55

The volumes of the containers are: V(large) = 2034.72 cm^3 V(small) = 678.24 cm^3

37. mathstudent55

The difference between the volumes is: 2034.72 cm^3 - 678.24 cm^3 = 1356.48 cm^3

38. mathstudent55

How did you get 452.16 cm^3?