To find the distance across the bottom of the aqueduct, we can use the formula for the cross-sectional area of a trapezoid. The formula for the area \( A \) of a trapezoid is given by:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \( b_1 \) is the distance across the top, \( b_2 \) is the distance across the bottom, and \( h \) is the depth. We can rearrange this formula to solve for \( b_2 \):
\[ b_2 = \frac{2A}{h} - b_1 \]
First, we need to calculate the cross-sectional area \( A \) using the flow rate and velocity. The flow rate \( Q \) is given by:
\[ Q = A \times v \]
Rearranging to solve for \( A \):
\[ A = \frac{Q}{v} \]
Given:
- Flow rate \( Q = 30,000 \) AFY (Acre-Feet per Year)
- Velocity \( v = 0.45 \) fps (feet per second)
- Distance across the top \( b_1 = 13 \) feet
- Depth \( h = 8 \) feet
We need to convert the flow rate from AFY to cubic feet per second (cfs) for consistency with the velocity units.
First, we need to convert the flow rate from acre-feet per year (AFY) to cubic feet per second (cfs). The conversion factors are:
- \( 1 \text{ year} = 31,536,000 \text{ seconds} \)
- \( 1 \text{ acre-foot} = 43,560 \text{ cubic feet} \)
Given:
\[ \text{Flow rate} = 30,000 \text{ AFY} \]
The flow rate in cubic feet per second is:
\[ \text{Flow rate} = \frac{30,000 \times 43,560}{31,536,000} \approx 41.4384 \text{ cfs} \]
The cross-sectional area \( A \) can be calculated using the flow rate and the velocity. Given:
\[ \text{Velocity} = 0.45 \text{ fps} \]
The area is:
\[ A = \frac{\text{Flow rate}}{\text{Velocity}} = \frac{41.4384}{0.45} \approx 92.0852 \text{ square feet} \]
Using the formula for the area of a trapezoid:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
We can solve for \( b_2 \):
\[ b_2 = \frac{2A}{h} - b_1 \]
Given:
- \( b_1 = 13 \text{ feet} \)
- \( h = 8 \text{ feet} \)
Substituting the values:
\[ b_2 = \frac{2 \times 92.0852}{8} - 13 \approx 10.0213 \text{ feet} \]
\(\boxed{b_2 \approx 10.0213 \text{ feet}}\)
Answered
