The base of a solid is the ellipse 9x2 + 25y2 = 225. Find the volume of the solid given that the cross sections perpendicular to the x-axis are equilateral triangles.

Question

turingtest · Answer

this is going to be very much the same thing
we need the same expression for volume$$V=\int A(x)dx$$so we need an expression for A(x)

always nice to have a drawing in front of you in these cases|dw:1331262302695:dw|remember that we can always write the function in terms of y=f(x)
now we have to imagine an equilateral triangle popping out of the ellipse
first, what will be the length of one side of the triangle?

turingtest · Answer

|dw:1331262583558:dw|this black line will be the length of one side of the triangle
how can we represent its length L in terms of x ?

anonymous · Answer

im not sure..im not too familar with elipses

turingtest · Answer

you don't have to be, you just need to visualize the triangles|dw:1331262772907:dw|here are four drawn in randomly (imagine their heights popping out of the page towards you)
The length of their sides L is a function of where you are on the graph

turingtest · Answer

now clearly one length we can identify in terms of x is f(x)
let us draw in a few random f(x)|dw:1331262949377:dw|ok, not so randomly drawn in, but it illustrates the point this way:
the length of each side of the triangle L is representable from the picture in terms of f(x)
what is L in terms of f(x) ?