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Tue 20 Feb, 2007 05:47 am
Hello!!
Problem:
A solid has a circular base of radius 4 units. Find the volume of the solid if every plane
section perpendicular to a fixed diameter is an equilateral triangle.
Solution:
Draw a circle on a three dimensional coordinate system, using the x-axis as the diameter
of the circle. The circle will intersect the x-axis and the y-axis at -4 and 4. The equation
of the circle is x2 + y2 = 16. Draw a representative cross section as an equilateral triangle
parallel to the z-axis. This cross-section has sides equal to 2y and area A = \/3 y2.
(The height of the triangle is \/3 y, using the Pythagorean Theorem.) Since y = \/(16 - x2),
then A = (\/3) (16 - x2). The differential (or width of the triangular prism) is dx.
V(x)=sqrt(3)*int_-4^4(16-x^2)dx=sqrt(3)*(16*x-x^3/3)= 256*sqrt(3)/3.
My question is can you explain to me how did they get side of 2y , I´know this is Pythagorean T. but still do not know how?!.Please help me!!
Draw an equilateral trangle in the y-z plane with its horizontal apex on the z axis.
The top half of the triangle is a 30,60,90 triangle whose sides are in the ratio 1:2: sqrt 3 (by pythagoras) so if the "side opposite" to 30 is y, the hypotenuse is 2y. (This is where sin30 = 1/2 comes from)
Area =1/2 base . height
(using "base"=2y and "height" in the z direction =y.sqrt3)
Area = (y^2).sqrt3
Hello!
Thank you so much for your time!!
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