Question 1188293: The direction of a railroad cut is perpendicular to the horizontal ridge AB. Assuming the sides of the ridge to be plane surfaces making equal angles with the horizontal, find the amount of earth removed if the base is a rectangle, 30 ft. wide by 400 ft. long, in a plane 60 ft. below the line AB; and the sides of the cut are inclined 45 degrees to the horizontal.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the volume of earth removed:
**1. Visualize the Problem:**
Imagine a horizontal ridge AB. The railroad cut goes straight down, perpendicular to AB. The sides of the cut slope inwards at a 45-degree angle. The base of the cut is a rectangle.
**2. Cross-Sectional Area:**
The key is to find the cross-sectional area of the cut. Because the sides are inclined at 45 degrees, and the cut is perpendicular to the ridge, the cross-section will be a trapezoid.
* **Base of trapezoid:** This is the width of the rectangular base of the cut, which is 30 ft.
* **Height of trapezoid:** This is the vertical depth of the cut, which is 60 ft.
* **Top of trapezoid:** Because the sides are at 45 degrees, the top of the trapezoid is wider than the base. For each side, the additional width is equal to the vertical depth (60 ft). Since there are two sides, the top of the trapezoid is 30 ft + 2 * 60 ft = 150 ft.
**3. Calculate the Area of the Trapezoid:**
Area = (1/2) * (base1 + base2) * height
Area = (1/2) * (30 ft + 150 ft) * 60 ft
Area = (1/2) * 180 ft * 60 ft
Area = 5400 sq ft
**4. Calculate the Volume:**
Volume = Area of cross-section * length
Volume = 5400 sq ft * 400 ft
Volume = 2,160,000 cu ft
**Answer:**
The amount of earth removed is 2,160,000 cubic feet.
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