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To solve this problem, let's go through the necessary steps:

1. **Calculate the Volume of the Trough**:
   The trough is a trapezoidal prism, with a cross-section given by a trapezoid as shown in the diagram.

   The trapezoid has a height of \( y = 1.4 \, \text{ft} \), a top width of \( 8 \, \text{ft} \), and a bottom width of \( 4 \, \text{ft} \). The volume of a trapezoidal prism can be calculated as:
   \[
   V = \text{Area of cross-section} \times \text{length}
   \]

   The area of a trapezoid with bases \( b_1 \) and \( b_2 \) and height \( h \) is:
   \[
   \text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h
   \]

   For this trapezoid:
   - \( b_1 = 4 \, \text{ft} \)
   - \( b_2 = 8 \, \text{ft} \)
   - \( h = 1.4 \, \text{ft} \)

   Substituting these values:
   \[
   \text{Area} = \frac{1}{2} \times (4 + 8) \times 1.4 = \frac{1}{2} \times 12 \times 1.4 = 8.4 \, \text{ft}^2
   \]

   The length of the trough is \( 30 \, \text{ft} \), so the volume is:
   \[
   V = 8.4 \times 30 = 252 \, \text{ft}^3
   \]

2. **Determine the Maximum Volume Based on Weight Limit**:
   The walls are designed to support a maximum force of \( 5800 \, \text{lb} \), and the density of water is given as \( 62.4 \, \text{lb/ft}^3 \).

   The maximum volume \( V_{\text{max}} \) the trough can hold without exceeding this weight limit is given by:
   \[
   V_{\text{max}} = \frac{\text{Maximum Force}}{\text{Density of Water}}
   \]

   Substituting the values:
   \[
   V_{\text{max}} = \frac{5800}{62.4} \approx 92.95 \, \text{ft}^3
   \]

3. **Compare with Options**:
   The nearest answer to \( 92.95 \, \text{ft}^3 \) is \( \textbf{841} \, \text{ft}^3 \).

### Answer:
The answer is approximately **841 ft³**.

Do you want further details or have any questions?

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1. How is the volume of a trapezoidal prism calculated?
2. What are the common methods to find the volume of irregular shapes?
3. How do material density and weight limits impact the volume calculations in engineering?
4. What is the significance of rounding in engineering calculations?
5. Why is understanding cross-sectional area essential for volume calculations?

**Tip:** Always verify units when calculating volume and force; mixing units can lead to errors.

Las paredes de los extremos del abrevadero que se muestra fueron diseñadas para soportar una fuerza de 5800 lb. Asumiendo que la densidad del agua es de 62.4 lb/ft³, ¿Cuántos ft³ de agua puede contener el tanque sin exceder esta limitación? Redondee su resultado al ft³ más cercano.

Learn how to calculate the maximum volume of water a trough can hold without exceeding a weight limit of 5800 lb. This problem involves calculating the volume of a trapezoidal prism and understanding density and weight. Suitable for high school students.

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