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What is the volume  of this shape in cubic yards

The shape shown in the image is a triangular prism. To find its volume, we can use the formula for the volume of a triangular prism:

\[
\text{Volume} = \text{Base Area} \times \text{Height of the prism}
\]

In this case, the base of the prism is a right triangle, and the height of the prism is the width \( w \).

1. **Step 1: Calculate the height \( h \) of the triangular base**:
   We are given the two sides of the triangle: 85 feet (the hypotenuse) and 56 feet (one of the legs). We can use trigonometry to find the height \( h \). Given that \( \theta = 26^\circ \), we can use the sine function:
   
   \[
   \sin(\theta) = \frac{h}{85}
   \]
   
   Solving for \( h \):
   
   \[
   h = 85 \times \sin(26^\circ)
   \]

2. **Step 2: Calculate the base of the triangular face**:
   The base of the triangle is the horizontal leg of the right triangle, which can be calculated using the cosine function:
   
   \[
   \cos(\theta) = \frac{\text{base}}{85}
   \]
   
   Solving for the base:
   
   \[
   \text{base} = 85 \times \cos(26^\circ)
   \]

3. **Step 3: Calculate the area of the triangular base**:
   The area of the triangular base is:
   
   \[
   \text{Area of base} = \frac{1}{2} \times \text{base} \times h
   \]

4. **Step 4: Calculate the volume**:
   Once we have the area of the base, we can multiply it by the width \( w = 56 \, \text{ft} \) to get the volume of the prism in cubic feet.

5. **Step 5: Convert to cubic yards**:
   Since 1 cubic yard = 27 cubic feet, we divide the volume in cubic feet by 27 to get the volume in cubic yards.

Let me calculate the steps for you.The volume of the triangular prism is approximately **2,952.12 cubic yards**.

Here are the intermediate results:
- Height (\( h \)) of the triangular base: \( 37.26 \, \text{ft} \)
- Base of the triangular face: \( 76.40 \, \text{ft} \)
- Volume in cubic feet: \( 79,707.29 \, \text{ft}^3 \)

Would you like more details, or do you have any questions about these calculations?

Here are 5 follow-up questions to consider:
1. How would the volume change if the angle \( \theta \) was increased?
2. What is the effect of increasing the width (\( w \)) on the total volume?
3. How would you calculate the surface area of the triangular prism?
4. Can the method used for this triangular prism be applied to other types of prisms?
5. How does the volume in cubic feet compare to the volume in cubic meters?

**Tip**: Always ensure that all units are consistent when working with volumes, especially when converting between cubic feet and cubic yards!

What is the volume of this shape in cubic yards?

Learn how to calculate the volume of a triangular prism with a base angle of 26 degrees. This problem covers the use of trigonometry and geometry to find the area of a triangular base and calculate the total volume in cubic yards. Suitable for high school students.

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