Trapezoidal Prism Reservoir: Calculate Water Volume (Question 29)

Hey guys! Ever wondered how much water a prism-shaped reservoir can hold? Let's dive into Question 29, which challenges us to calculate the volume of water in a right prism reservoir with a trapezoidal base. This is a classic geometry problem that combines our understanding of prisms and trapezoids. So, grab your thinking caps, and let's break it down step-by-step!

Understanding the Problem: The Trapezoidal Prism

First, let's visualize what we're dealing with. We have a reservoir shaped like a right prism. This means the sides are perpendicular to the base, which is a trapezoid. Now, a trapezoid is a quadrilateral with at least one pair of parallel sides. In our case, the trapezoidal base has two parallel sides (bases) measuring 6 meters and 10 meters. The height of this trapezoid (the perpendicular distance between the parallel sides) is 4 meters. The prism itself has a height, which is the distance between the two trapezoidal bases. This height is crucial for calculating the overall volume.

The key to solving this problem lies in understanding how to find the volume of a prism. Remember, the volume of any prism is simply the area of its base multiplied by its height. So, our main task is to figure out the area of the trapezoidal base and then multiply it by the height of the prism. Easy peasy, right? But before we jump into the calculations, let’s take a closer look at the properties of trapezoids and prisms to make sure we're all on the same page. Understanding these fundamentals will make the problem much clearer and prevent any common mistakes. Think of it like building a house – you need a solid foundation before you can start putting up the walls!

Calculating the Trapezoidal Base Area

The most important step in figuring out the reservoir's water capacity is calculating the area of the trapezoidal base. The formula for the area of a trapezoid is: Area = (1/2) * (base1 + base2) * height. Here, 'base1' and 'base2' are the lengths of the parallel sides of the trapezoid, and 'height' is the perpendicular distance between them. In our problem, we have base1 = 6 meters, base2 = 10 meters, and height = 4 meters. Let's plug these values into the formula: Area = (1/2) * (6 + 10) * 4. Following the order of operations, we first add the bases: 6 + 10 = 16. Then, we multiply by the height: 16 * 4 = 64. Finally, we multiply by 1/2 (or divide by 2): 64 / 2 = 32. So, the area of the trapezoidal base is 32 square meters.

It's super important to remember the units here! We're dealing with area, so the units are square meters. Forgetting the units is a common mistake, and it can lead to errors in the final answer. Now that we've got the base area, we're one step closer to finding the volume. Think of this as laying the foundation for our volume calculation. Without the correct base area, the final result won't be accurate. So, let's make sure we're crystal clear on this step before moving on to the next part of the problem. Understanding the area of the base is crucial, as it directly impacts the total volume the reservoir can hold. We've conquered the trapezoid, now let's tackle the prism!

Determining the Prism's Volume

Now that we've successfully calculated the area of the trapezoidal base (32 square meters), we can move on to the final step: finding the volume of the prism. As we discussed earlier, the volume of a prism is calculated by multiplying the area of its base by its height. The problem states that the height of the prism (the distance between the two trapezoidal bases) is also given. Let's assume for this example that the height of the prism is 5 meters (this value might be in the original question, which isn't fully provided in the context). So, to find the volume, we simply multiply the base area (32 square meters) by the prism's height (5 meters): Volume = 32 * 5. Performing this multiplication, we get a volume of 160 cubic meters.

Again, let's pay close attention to the units! Since we're calculating volume, the units are cubic meters (m³). This makes sense because volume represents the three-dimensional space occupied by the reservoir. So, the reservoir can hold 160 cubic meters of water. This calculation showcases the direct relationship between the base area and the overall volume. A larger base area, or a greater prism height, will result in a larger volume. It's like imagining filling the reservoir with tiny cubes – the more cubes that fit inside, the larger the volume. We've successfully navigated through the steps and arrived at the solution. High five!

Putting It All Together

Okay, guys, let's recap what we've done. We started with a problem asking us to find the volume of water in a prism-shaped reservoir with a trapezoidal base. We broke down the problem into smaller, manageable steps. First, we understood the geometry of the trapezoid and the prism. Then, we calculated the area of the trapezoidal base using the formula Area = (1/2) * (base1 + base2) * height. We found the base area to be 32 square meters. Next, we multiplied the base area by the height of the prism (assuming 5 meters for this example) to find the volume, which turned out to be 160 cubic meters.

This problem highlights the importance of understanding geometric formulas and applying them systematically. It also emphasizes the need to pay attention to units – using the correct units (square meters for area and cubic meters for volume) ensures the accuracy of our answer. Remember, solving geometry problems is like building a puzzle – each piece (formula, measurement, concept) fits together to form the complete solution. By breaking down complex problems into smaller steps, we can tackle even the trickiest challenges. So, next time you encounter a problem involving prisms and trapezoids, remember the steps we've covered here, and you'll be well on your way to success!

In conclusion, by applying the formulas for the area of a trapezoid and the volume of a prism, we successfully calculated the amount of water the reservoir can hold. This exercise demonstrates how understanding basic geometric principles can help us solve real-world problems. Keep practicing, guys, and you'll become geometry masters in no time!