Question

Sandra has a trough that will hold water for her cattle. It is 11 feet long, and each end is a trapezoid with a bottom measure of 4 feet and a top measure of 6 feet. If the trough is 2 feet deep, how many cubic feet of water will it hold? The area of a trapezoid is found using \(\frac{b_1+b_2}{2} h\), where \(b_1\) and \(b_2\) are the lengths of the two bases and \(h\) is the height. A. 55 cubic feet B. 110 cubic feet C. 220 cubic feet D. 20 cubic feet

Short answer

110 cubic feet. The correct answer is B.

Step-by-step solution

- Understand the problem

The trough is 11 feet long and each end forms a trapezoid. We need to find the volume of the trough. To do this, we'll first find the area of one of the trapezoidal ends and then multiply by the length of the trough.

- Recall the formula for the area of a trapezoid

The area of a trapezoid is given by \[A = \frac{b_1 + b_2}{2} h \] where \(b_1\) and \(b_2\) are the lengths of the two bases and \(h\) is the height.

- Identify the bases and height of the trapezoid

In this problem, the bottom base \(b_1\) is 4 feet, the top base \(b_2\) is 6 feet, and the height \(h\) of the trapezoid is the depth of the trough which is 2 feet.

- Calculate the area of one trapezoidal end

Using the values identified: \[A = \frac{4 + 6}{2} \times 2 \] Simplifying this: \[A = \frac{10}{2} \times 2 = 5 \times 2 = 10 \text{ square feet} \]

- Calculate the volume of the trough

Now, multiply the area of the trapezoidal end by the length of the trough to find the volume: \[V = A \times \text{length} = 10 \times 11 = 110 \text{ cubic feet} \]

- Choose the correct answer

The volume of the trough is 110 cubic feet, so the correct answer is B.

Key concepts

trapezoidal area formula

Understanding the area of a trapezoid is crucial for solving many geometry problems. A trapezoid is a four-sided figure with one pair of parallel sides, often referred to as bases. To find the area, we use the formula: \[A = \frac{b_1 + b_2}{2} h \] where \(b_1 \) and \(b_2\) are the lengths of the bases and \( h \) is the height. The formula helps you calculate the average length of the bases and then multiply it by the height.

For instance, let's apply this formula to a trapezoid with bases measuring 4 feet and 6 feet respectively, and a height of 2 feet. First, we find the average of the bases: \[ \frac{4 + 6}{2} = 5 \text{ feet} \]

We then multiply this average by the height: \[ 5 \times 2 = 10 \text{ square feet} \]

This means the area of the trapezoid is 10 square feet. This formula is commonly used not just in academic problems but also in real-life applications like calculating the area of sloped surfaces.

volume calculation

Calculating volume is essential when dealing with three-dimensional shapes. To find the volume of a trough, we need to consider the area of its cross-sectional shape and then multiply by its length. In Sandra's case, the cross-sectional shape is a trapezoid.

Once you've determined the area of the trapezoidal end (as described in the previous section), you can find the total volume by multiplying this area by the length of the trough. The steps are:

• Calculate the area of the cross-section (trapezoid)
• Multiply by the length of the trough

Using Sandra's trough as an example:

We've found the area of one trapezoidal end to be 10 square feet. The length of the trough is given as 11 feet. Thus, the volume is: \[ 10 \times 11 = 110 \text{ cubic feet} \]

This means Sandra's trough can hold 110 cubic feet of water. This method can be applied to any prismatic shape, where you multiply the area of the base by the height or length.

word problem solving

Word problems can sometimes seem daunting, but breaking them down into smaller steps makes them manageable. Here are general strategies to tackle word problems effectively:

• Read the problem carefully and identify what you're asked to find.
• Note down all the given information.
• Draw a diagram if possible to visualize the problem.
• Write down the relevant formulas and understand how they apply to the problem.
• Perform the calculations step by step.
• Double-check your work to ensure you've answered the question correctly.

Let's apply these strategies to Sandra's trough problem:

1. Read and understand: We need to find the volume of water the trough can hold.

2. Identify given information:

• Length of trough: 11 feet
• Bases of trapezoidal end: 4 feet and 6 feet
• Height (depth) of trapezoid: 2 feet

3. Visualize: Imagine the trough as a long container with trapezoidal ends.

4. Apply the formula: Use the trapezoid area formula first, then calculate the volume.

5. Perform calculations: As detailed before, the area is 10 square feet, and multiplying by the length (11 feet) gives 110 cubic feet.

6. Verify: Cross-check each step to ensure accuracy.

By following these methods, word problems can be less intimidating and more structured, helping you arrive at the correct solution efficiently.