how many cubic yards

Question

how many cubic yards of earth must be used to make a railroad embankment 500 feet long if a cross section of the embankment is an isosceles trapezoid whose bases are 20 feet and 45 feet whose legs are each 18 feet

Answer 1

Hi John,

 

If you divide the trapezoid into a central rectangle and two right triangles on the sides, you can calculate the areas of each of those figures and then multiply by 500 to get the volume.

 

So, the central rectangle is 20 x 20 --> 400 sq feet.  Multiplying by 500 --> 200000 cubic feet

 

Each right triangle has a hypotenuse of 18 and a base of 12.5 (12.5 + 12.5 = 20 = 45 which is the length of the base of the trapezoid).

 

Calculate the triangle height by the Pythagorean theorem:  12.5^2 + height^2 = 18^2

 

Height = 12.95 feet

 

Area of a triangle = 1/2 base x height.  Since there are two equal triangles, we'll leave out the 1/2 to get the area of both triangles = 12.5 x 12.95 = 161.88 sq feet.  Multiplying by 500 --> 80937.5 cubic feet

 

Adding the two volumes --> 200000 + 80937.5 = 280937.5 cubic feet

 

That's a lot of earth.

 

But we're not done because the question asks how many cubic yards of earth. 

 

1 cubic yard is 3 feet by 3 feet by 3 feet or 27 cubic feet.  So you have to divide the answer by 27 and you get. . . drum roll . . . . 10405 cubic yards.

 

Dan