https://file.aiquickdraw.com/mathbot/file/17265401736746tllyojm.jpg

To calculate the volume of the irregular-shaped hole shown in the image, we can break it down into simpler geometric shapes and sum their volumes. From the provided views (Plan View, Section View, and Isometric View), we can identify that the structure can be divided into:

1. A rectangular prism (flat-top section).
2. A triangular prism (sloped section).

### 1. **Rectangular Prism Volume (Flat-Top Section)**:
This part is a rectangular block. The dimensions from the **Plan View** and **Section View** are:
- Length (in the Plan View): 3000 mm
- Width: 2000 mm
- Height (in the Section View): 2500 mm

So, the volume of the rectangular prism is:
\[
V_1 = \text{Length} \times \text{Width} \times \text{Height} = 3000 \, \text{mm} \times 2000 \, \text{mm} \times 2500 \, \text{mm}
\]

### 2. **Triangular Prism Volume (Sloped Section)**:
The second part is a triangular prism with a sloped face. From the **Section View**, we see the following:
- Base width: 1000 mm (horizontal distance)
- Height: 2500 mm (vertical height)
- Depth (into the page, based on the Plan View): 5000 mm

So, the volume of the triangular prism is:
\[
V_2 = \frac{1}{2} \times \text{Base} \times \text{Height} \times \text{Depth} = \frac{1}{2} \times 1000 \, \text{mm} \times 2500 \, \text{mm} \times 5000 \, \text{mm}
\]

### Total Volume:
The total volume of the hole is the sum of the volumes of both shapes:
\[
V_{\text{total}} = V_1 + V_2
\]

Let me calculate these values.The total volume of the irregular-shaped hole is **21,250,000,000 mm³**. Converting this to cubic meters (since there are \(1,000,000 \, \text{mm}^3\) in a cubic meter), the volume is:

\[
V_{\text{total}} = 21.25 \, \text{m}^3
\]

This is the volume of soil required to fill the hole.

Would you like further details or have any questions?

Here are some related questions for further practice:
1. How would the volume change if the slope was steeper?
2. What if the width of the rectangular prism was increased by 500 mm?
3. How do you calculate the surface area of the entire excavated hole?
4. How would the volume calculation change if the height of the sloped section was reduced?
5. How do different geometric shapes affect the excavation cost?

**Tip:** Always double-check unit conversions when dealing with large-scale measurements like volume!

You were tasked to fill the currently excavated hole in the middle of your client's site. Compute for the volume of the soil needed in order to fill the irregular-shaped hole.

Calculate the volume of soil needed to fill an irregular-shaped hole. This problem involves breaking down the structure into a rectangular prism and a triangular prism, using volume formulas for each shape. Suitable for students in Grades 10-12.

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