Vid said:Think of the circles x^2+y^2 = (1.75)^2 z=0 and x^2+y^2 = (1.25)^2 z=6. Left y = 0. This gives us two points from our equations (1.75, 0, 0) and (1.25, 0 ,6). Now we make a parametric equations for the line going through these two points. v = (-.5,0,6)
r = Po + t*v
x = 1.75 - .5t
y = 0
z = 6t
We want x and y to be zero so we can solve for z. 1.75 = .5t t = 3.5
z = 6*(3.5) = 21. So the total height of the cone if it weren't missing anything would be 21. Find the volume for that cone then subtract out the volume of the cone with height 14 and base 1.25.
Vid said:I just realized a much simpler solution using similar triangles.
6/.5 = x/1.75 x = 12*1.75 = 21.
A frustum, also known as a truncated cone, is a three-dimensional shape that is formed when the top of a cone is cut off by a plane parallel to its base. It has two circular bases of different sizes connected by a curved surface.
The formula for calculating the volume of a frustum is V = (1/3) * π * h * (r12 + r22 + r1 * r2), where h is the height of the frustum, r1 is the radius of the larger base, and r2 is the radius of the smaller base.
Frustums are commonly used in architecture and engineering, such as in the design of buildings with sloped roofs or in the construction of cooling towers. They are also used in manufacturing of objects with tapered shapes, such as lampshades or traffic cones.
No, the volume of a frustum cannot be negative as it is a measure of the amount of space occupied by the shape. A negative volume would indicate that the shape does not exist, which is not possible.
The volume of a frustum is always less than the volume of a regular cone with the same base and height. This is because the frustum is missing a portion of the cone's volume due to the truncation of its top. The larger the difference in the radii of the bases, the greater the difference in volumes between the two shapes.