The problem describes Dominique's task to construct a water trough in the shape of a right trapezoidal prism. The given details include:

• The desired volume of the trough: $30 \, \text{ft}^3$
• The trapezoidal cross-section of the prism:
  • Height of the trapezoid: $1 \, \text{ft}$
  • Bases of the trapezoid: $2 \, \text{ft}$ (top) and $1 \, \text{ft}$ (bottom)

To find the length of the trough ($L$) required to achieve the volume of $30 \, \text{ft}^3$, we proceed as follows:

Step 1: Area of the Trapezoidal Cross-Section

The area $A$ of a trapezoid is calculated using: $A = \frac{1}{2} \times (b_1 + b_2) \times h$ where:

• $b_1 = 2 \, \text{ft}$ (top base),
• $b_2 = 1 \, \text{ft}$ (bottom base),
• $h = 1 \, \text{ft}$ (height).

Substitute these values: $A = \frac{1}{2} \times (2 + 1) \times 1 = \frac{1}{2} \times 3 \times 1 = 1.5 \, \text{ft}^2$

Step 2: Volume of the Prism

The volume $V$ of a prism is given by: $V = A \times L$ Since we want the volume $V = 30 \, \text{ft}^3$, we substitute $A = 1.5 \, \text{ft}^2$: $30 = 1.5 \times L$ Solving for $L$: $L = \frac{30}{1.5} = 20 \, \text{ft}$

Answer

Dominique should make the trough 20 feet long.

Would you like further explanation on any step, or do you have additional questions?

Here are some related questions you might consider:

1. How would the length change if the volume was doubled?
2. What if the bottom base of the trapezoid was increased to 2 feet?
3. How does changing the height of the trapezoid affect the length?
4. What would be the length if the volume needed was 45 cubic feet?
5. How can you derive the formula for the volume of a trapezoidal prism?

Tip: When working with prisms, always calculate the area of the base shape first; this area multiplied by the prism's length gives you the volume.