Embankment

The railway embankment 300 m long has a cross section of an isosceles trapezoid with bases of 14 m and 8 m. The trapezoidal arms are 5 m long. Calculate how much m3 of soil is in the embankment?

Result

V =  13200 m3

Solution:

l=300 m a=14 m c=8 m r=5 m  x=a−c2=14−82=3 m r2=h2+x2 h=r2−x2=52−32=4 m S=a+c2⋅ h=14+82⋅ 4=44 m2 V=l⋅ S=300⋅ 44=13200 m3l=300 \ \text{m} \ \\ a=14 \ \text{m} \ \\ c=8 \ \text{m} \ \\ r=5 \ \text{m} \ \\ \ \\ x=\dfrac{ a-c }{ 2 }=\dfrac{ 14-8 }{ 2 }=3 \ \text{m} \ \\ r^2=h^2+x^2 \ \\ h=\sqrt{ r^2-x^2 }=\sqrt{ 5^2-3^2 }=4 \ \text{m} \ \\ S=\dfrac{ a+c }{ 2 } \cdot \ h=\dfrac{ 14+8 }{ 2 } \cdot \ 4=44 \ \text{m}^2 \ \\ V=l \cdot \ S=300 \cdot \ 44=13200 \ \text{m}^3

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