How many cubic yards of earth must be used to make a railroad embankment 500 ft. long if a cross section of the embankment is an isosceles trapezoid whose bases are 20 ft. and 45 ft. and whose legs are 18 ft. each?
A. 42,088.05
B. 64,141.35
C. 70,144.98
D. 210,437.50

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:
\[
\text{Volume in cubic yards} = \frac{210,437.5}{27}
\]
\[
\approx 7,794.91 \text{ cubic yards}
\]

Based on the choices provided, none match 7,794.91 cubic yards

Determine the volume in cubic yards for a 500 ft. railroad embankment with an isosceles trapezoid cross section. This geometry problem involves calculating area, using the Pythagorean theorem for height, and converting cubic feet to cubic yards. Ideal for high school students studying volume and geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation