Question

A \(1.25\) -mile section of trapezoidal shaped aqueduct needs to be drained. The aqueduct is 5 feet wide at the base and 10 feet wide at the water line. If there is 9 -acre feet of water in the aqueduct, what is the depth?

Short answer

The depth of the water is approximately 7.92 feet.

Step-by-step solution

Understand the Shape of the Aqueduct

The aqueduct is trapezoidal in shape. This means the cross-sectional area of the water in the aqueduct forms a trapezoid. The trapezoid's bases are 5 feet and 10 feet wide.

Formula for the Area of a Trapezoid

The area of a trapezoid is given by the formula: \[ A =\frac{1}{2} (b_1 + b_2) h \] Here, \(b_1\) is 5 feet, \(b_2\) is 10 feet, and \(h\) is the depth.

Find the Cross-Sectional Area

Substitute the known values into the trapezoid area formula to get: \[ A = \frac{1}{2} (5 + 10) h = \frac{15}{2} h \]

Convert Acre-Feet to Cubic Feet

1 acre-foot is equal to 43,560 cubic feet. Therefore, 9 acre-feet is \( 9 \times 43,560 = 392,040 \) cubic feet.

Calculate Volume and Depth

Given the aqueduct length of 1.25 miles (1 mile = 5280 feet), the total length in feet is \( 1.25 \times 5280 = 6600 \) feet. The volume formula for a prism (extended trapezoid) is \[ V = A \times \text{length} \] Substituting the area and volume, we get: \[ 392,040 = \frac{15}{2} h \times 6600 \] Solving for \( h \): \[ 392,040 = 49500h \] \( h = \frac{392,040}{49,500} \approx 7.92 \text{ feet} \)

Key concepts

geometry of trapezoids

Understanding the geometry of trapezoids is crucial for solving problems related to shapes like aqueducts. A trapezoid is a 4-sided figure with at least one pair of parallel sides, known as its bases. Unlike rectangles and squares, trapezoids have slanted sides. Here are some key elements of trapezoid geometry:

• Two parallel sides (bases): In our problem, these are 5 feet and 10 feet wide.
• Two non-parallel sides (legs): These join the bases but are not equal in all trapezoidal shapes.
• The height (h): This is the perpendicular distance between the two bases.

Knowing these features lets us use the trapezoid area formula effectively: \( A = \frac{1}{2} (b_1 + b_2) h \). This formula gives the area of the trapezoid when you know the lengths of the bases and the height. For our aqueduct problem, we’re given the bases (5 feet and 10 feet) and need to find the height (or depth) to calculate the cross-sectional area.

volume conversion

To solve many real-world problems, it’s essential to convert measurements into compatible units. In this problem, we need to work with volume conversions. Specifically, we have the volume of water in the aqueduct given in acre-feet, which we need to convert to cubic feet. Here is the step-by-step process:

• Understand the unit: 1 acre-foot is the volume of water that covers one acre of area to a depth of one foot.
• Conversion factor: 1 acre-foot equals 43,560 cubic feet.
• Multiply to convert: To find the volume in cubic feet, multiply the given volume in acre-feet by 43,560.
  For example, 9 acre-feet is converted to cubic feet like this:
  
  \( 9 \times 43,560 = 392,040 \)

This conversion helps us relate the geometrical shape’s cross-sectional area to its total volume along the aqueduct’s length. Understanding these conversions helps you switch between units seamlessly while solving geometry-based volume problems.

area calculation

Calculating the area of a trapezoid is vital to solve for the volume in problems involving trapezoidal shapes. Here's how to calculate the cross-sectional area step-by-step:

• Identify the bases and the height: In our case, the bases are 5 feet and 10 feet, while the height is what we need to determine.
• Use the trapezoid area formula:
  
  \( A = \frac{1}{2} (b_1 + b_2) h \)
• Plug in the known values:
  
  \( A = \frac{1}{2} (5 + 10) h = \frac{15}{2} h \)
• Simplify: This simplifies to 7.5h, giving the area formula in terms of the unknown height.
• Calculate the volume using the length of the aqueduct: The formula for the volume (V) of a prism extended in length (l) is:
  
  \( V = A \times l \)
• Apply the given values: For the aqueduct length of 1.25 miles (which converts to 6600 feet), we set up the equation like this:
  
  \( 392,040 = \frac{15}{2} h \times 6600 \)
  Simplify and solve for h (depth):
  
  \( 392,040 = 49,500h \)
  \( h = \frac{392,040}{49,500} \approx 7.92 \text{ feet} \)

With these calculations, you understand how to determine the area and use it to solve for the volume, a crucial skill in geometry and practical applications.