To determine the volume of the railroad embankment, we can use the following approach:

Step 1: Calculate the Area of the Cross Section

Since the cross section is an isosceles trapezoid with bases $b_1 = 20$ feet and $b_2 = 45$ feet, and legs (sides) of $18$ feet each, we can calculate the area of this trapezoid.

The area $A$ of an isosceles trapezoid is given by: $A = \frac{1}{2} (b_1 + b_2) \cdot h$ where $h$ is the height of the trapezoid (the distance between the two bases).

To find $h$, we can use the Pythagorean theorem. The horizontal difference between the bases is $b_2 - b_1 = 45 - 20 = 25$ feet. Dividing this by 2 gives 12.5 feet for each half of the difference (forming two right triangles with legs 12.5 feet and height $h$).

Applying the Pythagorean theorem: $h = \sqrt{18^2 - 12.5^2}$ $h = \sqrt{324 - 156.25} = \sqrt{167.75} \approx 12.95 \text{ feet}$

Now, substitute $b_1$, $b_2$, and $h$ into the area formula: $A = \frac{1}{2} \times (20 + 45) \times 12.95$ $A = \frac{1}{2} \times 65 \times 12.95$ $A \approx 420.88 \text{ square feet}$

Step 2: Calculate the Volume of the Embankment

The volume $V$ of the embankment can be found by multiplying the area of the cross section by the length of the embankment: $V = A \times \text{length}$ $V = 420.88 \times 500$ $V \approx 210,437.5 \text{ cubic feet}$

Step 3: Convert Cubic Feet to Cubic Yards

Since there are 27 cubic feet in a cubic yard: $\text{Volume in cubic yards} = \frac{210,437.5}{27}$ $\approx 7,794.91 \text{ cubic yards}$

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