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Tuesday, December 18, 2012

Triangular Prism Problem

Category: Solid Geometry, Trigonometry

"Published in Newark, California, USA"

The Pennsylvania Railroad found it necessary, owing to land slides upon the roadbed, to reduce the angle of inclination of one bank of a certain railway cut near Pittsburgh, PA, from an original angle of 45º to a new angle of 30º. The bank as it originally stood was 200 ft. long and had a slant length of 60 ft. Find the amount of the earth removed, if the top level of the bank remained unchanged. 

Photo by Math Principles in Everyday Life


The given figure is the cross section of a bank. The darker area of the cross section will be removed so that the inclination will be 30º instead of 45º. The darker area is also an obtuse triangle and we need to find the area of that one. Let's start to get the sides of the triangle by analyzing the figure as follows

Photo by Math Principles in Everyday Life

Consider the 45º triangle in order to solve for h and c as follows

Consider the 30º triangle in order to solve for d as follows

The base of a triangle can be calculated as follows

The area of a triangle is also the area of the base is calculated as follows

Finally, the volume of a prism is also the amount of the soil or earth removed from the bank is calculated as follows