## anonymous one year ago The base of a solid in the xy-plane is the circle x2 + y2 = 16. Cross sections of the solid perpendicular to the y-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

1. DanJS

Do you know what the solid is?

2. anonymous

no I don't know how to start this problem

3. DanJS

|dw:1438722075391:dw|

4. DanJS

They gave you x^2 + y^2 = 16 = r^2 That is a circle radius 4 centered at the origin, in the XY-plane...

5. DanJS

you recognize that?

6. anonymous

yes

7. DanJS

|dw:1438722233544:dw|

8. DanJS

bad drawing, but that is a slice of the shape perpendicular to the Y axis

9. DanJS

saying it is equilateral, means that is a Right Circular Cone

10. anonymous

okay so you need the volume formula for a right circular cone correct?

11. DanJS

yes, just remember, it is the area of the base times 1/3 the height

12. anonymous

ok

13. DanJS

need to figure the height

14. DanJS

Loooking at the cross section at the XZ axis head on... the base is 2 times the radius

15. DanJS

|dw:1438722512573:dw|

16. DanJS

THe y axis is pointing out from the page, that is the origin

17. DanJS

Equilateral triangle, so the diagonals are the same as the base, 2 radiuses

18. DanJS

19. DanJS

r=4, use pythagorean theorem to figure the height

20. DanJS

|dw:1438722659628:dw|

21. DanJS

You get it all?

22. anonymous

yes so far

23. DanJS

|dw:1438722796113:dw|

24. DanJS

you can only put in the 2r because they told you the cross sections are equilateral

25. DanJS

|dw:1438722884153:dw|

26. DanJS

pelletty drawing, but hope you see where those measurments come from

27. anonymous

yes it is understood

28. DanJS

k, so you know the radius and height, you can calculate the volume $V = \frac{ 1 }{ 3 }\pi*r^2*h$

29. DanJS

r = 4 h = sqrt(8^2 - 4^2)

30. anonymous

The way you set it up is multiply 1/3 by pi(4)^2(sqrt(8^2-4^2))?

31. DanJS

yes, if you simplify the height first, it is... $h = \sqrt{64-16} = 4\sqrt{3}$

32. anonymous

okay thank you very much for the explanation

33. DanJS

$V = \frac{ 1 }{ 3 }*\pi*4^2*4\sqrt{3}$

34. DanJS

welcome, that was a fun prob.. hope ya get it