Math Is Fun Forum

leochai86

Member

Registered: 2008-02-10

Posts: 1

Anyone can help to solve this 2 question? Thank~

1)A cylindrical hole of radius "a" is bored through a solid right-circular cone of height "h" and base radius b>a.
   If the axis of the hole lies along that of the cone find the volume of the remaining part of the cone.

2)Find the
  (a)area
  (b)centre of mass

of the solid plate defind by the region in the xy-plane bounded by y=x^2, and y = x.
Assuming that the denstiy of the plate is proportional to the distance from the x-axis and the total mass of the region is M.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Anyone can help to solve this 2 question? Thank~

(1)
If the boring process is started from the base of the cone, a cylinder will be removed followed by a smaller cone on top. The height of the smaller cone can be found by similar triangles. Subtract this from h to get the height the cylindrical portion removed. Now you can find the volumes of the cylindrical and conical portions removed.

If I’ve done it correctly myself, the answer should be

(2)
(a) Integrate x−x[sup]2[/sup] from 0 to 1.
(b) First we find a formula for the density. Divide the plate into horizontal strips of thickness δx. The strip at horizontal distance x from O has density kx for some positive constant k, and its area is (x−x[sup]2[/sup])δx; ∴ its mass is k(x[sup]2[/sup]−x[sup]3[/sup])δx. We know that the total mass is M; hence

which gives us k = 6M.

So the mass of a strip at horizontal distance x from O is 6M(x[sup]2[/sup]−x[sup]3[/sup])δx.

Now the formula for the centre of mass is

(distance of centre of mass from O)×(total mass) = ∑[(mass of strip)×(distance of strip from O)] = ∑[6Mx(x[sup]2[/sup]−x[sup]3[/sup])δx]

Thus the distance of centre of mass from O of the plate is

dawg

Guest

Re: Anyone can help to solve this 2 question? Thank~

yah what she said