Find the volume of a trough 5 meters long whose ends are equilateral triangles, each of whose sides has a length of 2 meters .

Answer 1

The trough is created by moving the equilateral triangle perpendicular to the plane in which it is drawn. If the triangle is drawn on a standard x,y coordinate plane, the trough is created by translating the triangle in the "z" direction. Therefore, the volume of the trough is simply the area of the triangle multiplied by the length it is translated in the "z" direction, which is given as 5 meters.

The equilateral triangle has sides measuring 2 meters. If you draw a line bisecting one of the angles, you will wind up with a right triangle with hypotenuse 2 meters, short leg 1 meter, longer leg square root of 3 meters. The area of a right triangle is A = (1/2)*base*height so A= (1/2)*1*(square root 3) = (square root 3)/2

The equilateral triangle contains two of these right triangles, so the area of the equilateral triangle is simply (square root 3) square meters. Multiplying this area by the length of the trough (5 meters) gives the final volume of the trough as 5(square root 3) cubic meters.

Answer 2

The altitude of the equilateral triangle is 5√3/2

Area of triangular end = (1/2)(base)(height) = (1/2)(5)(5√3/2) = 25√3/4

Volume of trough = (area of triangular end)(height) = (25√3/4)(5) = 125√3 / 4