Calculate the volume of the earthwork of an embankment if the width of the formation level is 8 m, depth of cutting is 5 m, slopes at two sides are 1:2 and 1:3 and the length of the embankment is 250 m.

a.
21545 m3

b.
45255 m3

c.
25625 m3

d.
30565 m3

To calculate the volume of earthwork for an embankment, we can use the formula for the volume of a prismoidal shape, as the embankment consists of a trapezoidal cross-section that extends along a certain length.

Here are the steps:

### Step 1: Determine the Area of the Cross-section
The cross-section of the embankment is trapezoidal. The area of a trapezoid can be calculated as:

\[
A = \frac{1}{2} \times \left(B_1 + B_2\right) \times h
\]

Where:
- \(B_1\) is the width of the top (formation level) = 8 m,
- \(B_2\) is the width of the bottom, which we need to calculate considering the side slopes,
- \(h\) is the depth of cutting (height) = 5 m.

### Step 2: Calculate the Bottom Width
For each slope, we need to account for the horizontal distance that the slope adds on both sides:
- On one side, the slope is 1:2, meaning the horizontal distance is \(2 \times 5 = 10 \text{ m}\),
- On the other side, the slope is 1:3, meaning the horizontal distance is \(3 \times 5 = 15 \text{ m}\).

Thus, the total width at the bottom is:
\[
B_2 = 8 + 10 + 15 = 33 \text{ m}.
\]

### Step 3: Calculate the Area of the Cross-section
Now, we can substitute the values into the trapezoidal area formula:

\[
A = \frac{1}{2} \times (8 + 33) \times 5 = \frac{1}{2} \times 41 \times 5 = 102.5 \text{ m}^2.
\]

### Step 4: Calculate the Volume
The total volume is the cross-sectional area multiplied by the length of the embankment (250 m):

\[
V = 102.5 \times 250 = 25,625 \text{ m}^3.
\]

Thus, the correct volume of the earthwork is:

\[
\boxed{25,625 \text{ m}^3}.
\]

### The correct answer is: c. 25,625 m³

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Would you like more details or have any questions?

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Here are 5 related questions:
1. How would the volume change if the formation level width increased to 10 m?
2. What if the depth of cutting was reduced to 4 m with the same slopes?
3. How would different slope ratios affect the volume calculation?
4. Can the same formula be used for irregular cross-sections? Why or why not?
5. How is the prismoidal method applied to more complex geometries?

**Tip**: Always verify the unit consistency in any geometric or earthwork problem. Mixing units can lead to errors!

Calculate the volume of the earthwork of an embankment if the width of the formation level is 8 m, depth of cutting is 5 m, slopes at two sides are 1:2 and 1:3 and the length of the embankment is 250 m.

Learn how to calculate the volume of earthwork for an embankment with a trapezoidal cross-section. This problem covers key concepts in geometry and volume calculation, including the area of trapezoids and the prismoidal method. Suitable for students in Grades 9-12.

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