The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square. The solid has volume:
(A)
(B)
(C)
(D)
(E)
.
The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square. The solid has volume:
(A)
(B)
(C)
(D)
(E)
.
Place the circle in a coordinate system with the center at the origin. The equation of the circle is
Each slice perpendicular to the x-axis can be approximated by a right rectangular prism with dimensions 2y by 2y by dx.
The volume V of the prism is
Integrate from -a to a to get the volume of the solid
Ione
I have been thinking about this for the past while. Basically, what you did is you put the the base in the z=0 plane, and then uintegrated over x, using with the volume of each slice being 2y by 2y by dx. Which as I see it is correct, provided that each slice is the same height. Each slice has to have the same height by symmetrtry, as It doesn't matter when you rotate the object. This gives the total volume as .
However, the volume of a cylinder is . Taking our h to be twice the radius (2r) then gives the volume as 2*pi*a^3, which means that it has two different volumes, or something that I am thinking of is wrong. Is the object described not a cylinder? Which actually doesn't change the problem, because when you use your method to find the cylinder volume, you get 23*y*hdx, which still does not give any pi dependance. So what am I missing?
It claims every plane perpendicular to a diameter is a square. This means that the height has to be the same at every point surely? By symmettry?
Consider first the diameter in the x direction. The plane perpendicular to that diameter is a square. Now consider the diameter in the y direction, the plane perpendicular to that diameter is a square. Now consider the diameter in the 45 degree direction, the plane perpendicular to that diameter is a square, furthermore, it passes through the two previous planes, hence the squares are all the same height. Wrong in some way?