Some more volumes questions (1 Viewer)

[Grey Council]

A closed vessel tapers to the points X,Y at its ends. The cross-section of the vessel by a plane perpendicular to XY, at a distance h cm from X is h^3(4-h)^2 cm squared. Find its volume.

I'm confused.

Prove by integration that the volume of a pyramid of which the base is any closed polygon is one-third of the area of the base times the perpendicular distance of the vertex from the plane of the base.



Find by integration the volume of a right cone with radius of base r and vertical angle 2@.

humph

 

[xiao1985]

uhm... took me a while, but finally did them all =)

... altho q3's answer doesn't seem to be too right @@

 

[CM_Tutor]

A closed vessel tapers to the points X,Y at its ends. The cross-section of the vessel by a plane perpendicular to XY, at a distance h cm from X is h^3(4-h)^2 cm squared. Find its volume.

I'm confused.

This question is not very clear, and to understand what it means, you have to realise that if it tapers to points, then the cross sectional area at the ends is zero. Thus one of X or Y is at h = 0, and the other at h = 4.

So, we have δV = h³(4 - h)² δh

So, V_solid = lim_δh-->0Σ (h=0-->h=4) δV = ∫ (0-->4) h³(4 - h)² dh

and the rest is just integration

 

[CM_Tutor]

Prove by integration that the volume of a pyramid of which the base is any closed polygon is one-third of the area of the base times the perpendicular distance of the vertex from the plane of the base.

To do this in any easy way, you need to recall a result from year 9 Maths. Similar figures with some equivalent distance in ratio a:b have areas in ratio a²:b²

So, take a pyramid of base area A and height H, and slice it at a height h above the base, with the slice of thickness δh and volume δV.

By similar figures, A_slice / A = [(H - h) / H]²

So, δV = (1 - h / H)²A δh

So, V_solid = lim_δh-->0 Σ (h=0-->h=H) δV = ∫ (0-->H) (1 - h / H)²A dh = AH / 3, as required.

 

[CM_Tutor]

For the third one, xiao1985's answer has the semi-vertical angle as 2@. It is the vertical angle that is 2@, and hence the semi-vertical angle is @.

Each slice should be a circle in cross-section, and the answer is:

V = (πr³cot @) / 3