1) Find the volume of the frustum of a right circular cone with height h = 134 ,
ID: 2848126 • Letter: 1
Question
1) Find the volume of the frustum of a right circular cone with height h=134, lower base radius R=25, and top radius r=15.
2) Find the volume of a cap of a sphere with radius r=72 and heght= 25
3) he base of a certain solid is an elliptical region with boundary curve
36x2+16y2=576. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
Use the formula V=?baA(x)dx to find the volume of the solid.
The lower limit of integration is a =
The upper limit of integration is b =
The base of the triangular cross-section is the following function of x:
The height of the triangular cross-section is the following function of x:
The area of the triangular cross-section is A(x)=
Thus the volume of the solid is V =
4) The base of a certain solid is the triangle with vertices (0,0), (2,0), and (0,3). Cross-sections perpendicular to the
Explanation / Answer
1)That's basically a cone cut off of a cone with the bottom section left over.
The volume of a cone or any other pyramid is the area of the base times the height divided by three.
Therefore the volume of this frustum is the volume of the big cone minus the volume of the little cone.
Let x = how high the converging point is above the frustum.
To find out how high the big cone goes you have to use some more in depth geometry, such as similar triangles. For example, take a look at this relation. x / (h + x) = r / R. It follows directly from similar triangles.
You can get x = r*h / (R - r) if you do the algebra which I don't want to do right now.
V = (1/3)pi*R^2*(h + x) - (1/3)pi*r^2*x
V = (1/3)pi*(h*R^2 + x(R^2 - r^2))
V = (1/3)pi*(h*R^2 + r*h*(R^2 - r^2) / (R - r))
V = (1/3)h*pi(R^2 + R*r + r^2) = (1/3)134*pi(25^2 + 25*15 + 15^2) = 171810.333 cm^3
2)Break up the sphere into elementary cylinders of radius x and height dy, obtained by sectioning the sphrere by planes perpendicular to one of its diameter, each one at a distance y from the sphere 's center. Then, each elementary cylinder has volume
dV = ? x
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